INTRODUCTION TO OPTICAL WAVEGUIDE ANALYSIS
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INTRODUCTION TO OPTICAL WAVEGUIDE ANALYSIS Solving Maxwell's Equations and the Schrodinger Equation
KENJI KAWANO and TSUTOMU KITOH
This book is printed on acid-free paper. (S) Copyright © 2001 by John Wiley & Sons. All rights reserved. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4744. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail:
[email protected]. For ordering and customer service, call 1-800-CALL-WILEY. Library of Congress Cataloging-in-Publication Data: Kawano, Kenji. [Hikari doharo kaiseki no kiso. English] Introduction to optical waveguide analysis: solving Maxwell's equations and the Schrodinger equation/by Kenji Kawano, Tsutomu Kitoh. p. cm. ISBN 0-471-40634-1 (cloth) 1. Optical waveguides—Mathematical models. 2. Maxwell equations. 3. Schrodinger equation. I. Kitoh, Tsutomu. II. Title. TAI1750.K3913 2001 621.381'331—dc21 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1
2001017920
To our wives, Mariko and Kumiko
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CONTENTS
Preface / xi
1
Fundamental Equations
1
1.1 Maxwell's Equations / 1 1.2 Wave Equations / 3 1.3 Poynting Vectors / 7 1.4 Boundary Conditions for Electromagnetic Fields / 9 Problems / 10 Reference / 12
2
Analytical Methods
13
2.1 Method for a Three-Layer Slab Optical Waveguide / 13 2.2 Effective Index Method / 20 2.3 Marcatili's Method / 23 2.4 Method for an Optical Fiber / 36 Problems / 55 References / 57 vii
Viii
CONTENTS
3 Finite-Element Methods
59
3.1 Variational Method / 59 3.2 Galerkin Method / 68 3.3 Area Coordinates and Triangular Elements / 72 3.4 Derivation of Eigenvalue Matrix Equations / 84 3.5 Matrix Elements / 89 3.6 Programming / 105 3.7 Boundary Conditions / 110 Problems / 113 References / 115 4 Finite-Difference Methods
117
4.1 Finite-Difference Approximations / 118 4.2 Wave Equations / 120 4.3 Finite-Difference Expressions of Wave Equations / 127 4.4 Programming / 150 4.5 Boundary Conditions / 153 4.6 Numerical Example / 160 Problems / 161 References / 164 5
Beam Propagation Methods
165
5.1 Fast Fourier Transform Beam Propagation Method / 165 5.2 Finite-Difference Beam Propagation Method / 180 5.3 Wide-Angle Analysis Using Fade Approximant Operators / 204 5.4 Three-Dimensional Semivectorial Analysis / 216 5.5 Three-Dimensional Fully Vectorial Analysis / 222 Problems / 227 References / 230 6
Finite-Difference Time-Domain Method 6.1 Discretization of Electromagnetic Fields / 233 6.2 Stability Condition / 239 6.3 Absorbing Boundary Conditions / 241
233
CONTENTS
IX
Problems / 245 References / 249 7 Schrodinger Equation
251
7.1 Time-Dependent State / 251 7.2 Finite-Difference Analysis of Time-Independent State / 253 7.3 Finite-Element Analysis of Time-Independent State / 254 References / 263 Appendix A Vectorial Formulas
265
Appendix B Integration Formula for Area Coordinates
267
Index
273
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PREFACE
This book was originally published in Japanese in October 1998 with the intention of providing a straightforward presentation of the sophisticated techniques used in optical waveguide analyses. Apparently, we were successful because the Japanese version has been well accepted by students in undergraduate, postgraduate, and Ph.D. courses as well as by researchers at universities and colleges and by researchers and engineers in the private sector of the optoelectronics field. Since we did not want to change the fundamental presentation of the original, this English version is, except for the newly added optical fiber analyses and problems, essentially a direct translation of the Japanese version. Optical waveguide devices already play important roles in telecommunications systems, and their importance will certainly grow in the future. People considering which computer programs to use when designing optical waveguide devices have two choices: develop their own or use those available on the market. A thorough understanding of optical waveguide analysis is, of course, indispensable if we are to develop our own programs. And computer-aided design (CAD) software for optical waveguides is available on the market. The CAD software can be used more effectively by designers who understand the features of each analysis method. Furthermore, an understanding of the wave equations and how they are solved helps us understand the optical waveguides themselves. Since each analysis method has its own features, different methods are required for different targets. Thus, several kinds of analysis methods have xi
Xli
PREFACE
to be mastered. Writing formal programs based on equations is risky unless one knows the approximations used in deriving those equations, the errors due to those approximations, and the stability of the solutions. Mastering several kinds of analysis techniques in a short time is difficult not only for beginners but also for busy researchers and engineers. Indeed, it was when we found ourselves devoting substantial effort to mastering various analysis techniques while at the same time designing, fabricating, and measuring optical waveguide devices that we saw the need for an easy-to-understand presentation of analysis techniques. This book is intended to guide the reader to a comprehensive understanding of optical waveguide analyses through self-study. It is important to note that the intermediate processes in the mathematical manipulations have not been omitted. The manipulations presented here are very detailed so that they can be easily understood by readers who are not familiar with them. Furthermore, the errors and stabilities of the solutions are discussed as clearly and concisely as possible. Someone using this book as a reference should be able to understand the papers in the field, develop programs, and even improve the conventional optical waveguide theories. Which optical waveguide analyses should be mastered is also an important consideration. Methods touted as superior have sometimes proven to be inadequate with regard to their accuracy, the stability of their solutions, and central processing unit (CPU) time they require. The methods discussed in this book are ones widely accepted around the world. Using them, we have developed programs we use on a daily basis in our laboratories and confirmed their accuracy, stability, and effectiveness in terms of CPU time. This book treats both analytical methods and numerical methods. Chapter 1 summarizes Maxwell's equations, vectorial wave equations, and the boundary conditions for electromagnetic fields. Chapter 2 discusses the analysis of a three-layer slab optical waveguide, the effective index method, Marcatili's method, and the analysis of an optical fiber. Chapter 3 explains the widely utilized scalar finite-element method. It first discusses its basic theory and then derives the matrix elements in the eigenvalue equation and explains how their calculation can be programmed. Chapter 4 discusses the semivectorial finite-difference method. It derives the fully vectorial and semivectorial wave equations, discusses their relations, and then derives explicit expressions for the quasi-TE and quasi-TM modes. It shows formulations of Ex and Hy expressions for the quasi-TE (transverse electric) mode and Ey and Hx expressions for the quasi-TM (transverse magnetic) mode. The none-
PREFACE
XiM
quidistant discretization scheme used in this chapter is more versatile than the equidistant discretization reported by Stern. The discretization errors due to these formulations are also discussed. Chapter 5 discusses beam propagation methods for the design of two- and three-dimensional (2D, 3D) optical waveguides. Discussed here are the fast Fourier transform beam propagation method (FFT-BPM), the finite-difference beam propagation method (FD-BPM), the transparent boundary conditions, the wideangle FD-BPM using the Fade approximant operators, the 3D semivectorial analysis based on the alternate-direction implicit method, and the fully vectorial analysis. The concepts of these methods are discussed in detail and their equations are derived. Also discussed are the error factors of the FFT-BPM, the physical meaning of the Fresnel equation, the problems with the wide-angle FFT-BPM, and the stability of the FD-BPM. Chapter 6 discusses the finite-difference time-domain method (FD-TDM). The FD-TDM is a little difficult to apply to 3D optical waveguides from the viewpoint of computer memory and CPU time, but it is an important analysis method and is applicable to 2D structures. Covered in this chapter are the Yee lattice, explicit 3D difference formulation, and absorbing boundary conditions. Quantum wells, which are indispensable in semiconductor optoelectronic devices, cannot be designed without solving the Schrodinger equation. Chapter 7 discusses how to solve the Schrodinger equation with the effective mass approximation. Since the structure of the Schrodinger equation is the same as that of the optical wave equation, the techniques to solve the optical wave equation can be used to solve the Schrodinger equation. Space is saved by including only a few examples in this book. The quasi-TEM and hybrid-mode analyses for the electrodes of microwave integrated circuits and optical devices have also been omitted because of space limitations. Finally, we should mention that readers are able to get information on the vendors that provide CAD software for the numerical methods discussed in this book from the Internet. We hope this book will help people who want to master optical waveguide analyses and will facilitate optoelectronics research and development.
KENJI KAWANO and TSUTOMU KITOH Kanagawa, Japan March 2001
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INTRODUCTION TO OPTICAL WAVEGUIDE ANALYSIS
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CHAPTER 1
FUNDAMENTAL EQUATIONS
This chapter summarizes Maxwell's equations, vectorial wave equations, and the boundary conditions for electromagnetic fields.
1.1 MAXWELL'S EQUATIONS The electric field E (in volts per meter), the magnetic field H (amperes per meter), the electric flux density D (coulombs for square meters), and the magnetic flux density B (amperes per square meter) are related to each other through the equations
where the permittivity e and permeability n are defined as
Here, e0 and /^0 are the permittivity and permeability of a vacuum, and er and \ir are the relative permittivity and permeability of the material. Since the relative permeability \ir is 1 for materials other than magnetic 1
2
FUNDAMENTAL EQUATIONS
materials, it is assumed throughout this book to be 1. Denoting the velocity of light in a vacuum as c0, we obtain
The current density J (in amperes per square meter) in a conductive material is given by
The electromagnetic fields satisfy the following well-known Maxwell equations [1]:
Since the equation V-(VxA) = 0 holds for an arbitrary vector A, from Eqs. (1.8) and (1.9), we can easily derive
The current density J is related to the charge density p (in coulombs per square meter) as follows:
Equations (1.10) and (1.11) can be derived from Eqs. (1.8), (1.9), and (1.12).
1.2 WAVE EQUATIONS
3
1.2 WAVE EQUATIONS Let us assume that an electromagnetic field oscillates at a single angular frequency co (in radians per meter). Vector A, which designates an electromagnetic field, is expressed as
Using this form of representation, we can write the following phasor expressions for the electric field E, the magnetic field H, the electric flux density D, and the magnetic flux density B:
In what follows, for simplicity we denote E, H, D, and B in the phasor representation as E, H, D, and B. Using these expressions, we can write Eqs. (1.8) to (1.11) as
where it is assumed that [ir = 1 and p — 0.
1.2.1 Wave Equation for Electric Field E Applying a vectorial rotation operator Vx to Eq. (1.18), we get
4
FUNDAMENTAL EQUATIONS
Using the vectorial formula
we can rewrite the left-hand side of Eq. (1.22) as
The symbol V2 is a Laplacian defined as
Since Eq. (1.21) can be rewritten as
we obtain
Thus, the left-hand side of Eq. (1.22) becomes
On the other hand, using Eq. (1.19), we get for the right-hand side of Eq. (1.22)
where k0 is the wave number in a vacuum and is expressed as
1.2
WAVE EQUATIONS
5
Thus, for a medium with the relative permittivity en the vectorial wave equation for the electric field E is
And using the wave number k in that medium, given by
we can rewrite Eq. (1.30) as
When the relative permittivity &r is constant in the medium, this vectorial wave equation can be reduced to the Helmholtz equation
1.2.2 Wave Equation for Magnetic Field H Applying the vectorial rotation operator Vx to Eq. (1.19), we get
Thus,
Using
6
FUNDAMENTAL EQUATIONS
obtained from Eqs. (1.19) and (1.20), we get from Eq. (1.30) the following vectorial wave equation for the magnetic field H:
Using Eq. (1.31), we can rewrite Eq. (1.36) as
When the relative permittivity &r is constant in the medium, this vectorial wave equation can be reduced to the Helmholtz equation
Now, we discuss an optical waveguide whose structure is uniform in the z direction. The derivative of an electromagnetic field with respect to the z coordinate is constant such that
where ft is the propagation constant and is the z-directed component of the wave number k. The ratio of the propagation constant in the medium, ft, to the wave number in a vacuum, k$, is called the effective index:
When i0 is the wavelength in a vacuum,
where Aeff = A0/neff is the z-directed component of the wavelength in the medium. The physical meaning of the propagation constant ft is the phase rotation per unit propagation distance. Thus, the effective index «eff can be interpreted as the ratio of a wavelength in the medium to the wavelength in a vacuum, or as the ratio of a phase rotation in the medium to the phase rotation in a vacuum.
1.3 POYNTING VECTORS
7
We can summarize the Helmholtz equation for the electric field E as
or
For the magnetic field H, on the other hand, we get the Helmholtz equation
or
where we used the definition V]_ = tf/dx2 + tf/dy2.
1.3 POYNTING VECTORS In this section, the time-dependent electric and magnetic fields are expressed as E(r, t) and H(r, t\_ and thejime-independent electric and magnetic fields are expressed as E(r) and H(r). Because the voltage is the integral of an electric field and because the magnetic field is created by a current, the product of the electric field and the magnetic field is related to the energy of the electromagnetic fields. Applying a divergence operator V • to ExH, we get
Substituting Maxwell's equations (1.8) and (1.9) into this equation, we get
8
FUNDAMENTAL EQUATIONS
When Eq. (1.46) is integrated over a volume V, we get
where we make use of Gauss's law and n designates a component normal to the surface S of the volume V. The first two terms of the last equation correspond to the rate of the reduction of the stored energy in volume V per unit time, while the third term corresponds to the rate of reduction of the energy due to Joule heating in volume V per unit time. Thus, the term J5(ExH)w dS is considered to be the rate of energy loss through the surface. Thus,
is the energy that passes through a unit area per unit time. It is called a Poynting vector. For an electromagnetic wave that oscillates at a single angular frequency CD, the time-averaged Poynting vector (S) is calculated as follows:
Here, we used (exp(/2<0f)) = (exp(—j2cot}) = 0. Thus, for an electromagnetic wave oscillating at a single angular frequency, the quantity
1.4
BOUNDARY CONDITIONS FOR ELECTROMAGNETIC FIELDS
9
is defined as a complex Poynting vector and the energy actually propagating is considered to be the real part of it.
1.4 BOUNDARY CONDITIONS FOR ELECTROMAGNETIC FIELDS The boundary conditions required for the electromagnetic fields are summarized as follows: (a) Tangential components of the electric fields are continuous such that
(b) When no current flows on the surface, tangential components of the magnetic fields are continuous such that
When a current flows on the surface, the magnetic fields are discontinuous and are related to the current density Js as follows:
Or, since the magnetic field and the current are perpendicular to each other, the vectorial representation is
(c) When there is no charge on the surface, the normal components of the electric flux densities are continuous such that
When there are charges on the surface, the electric flux densities are discontinuous and are related to the charge density ps as follows:
10
FUNDAMENTAL EQUATIONS
(d) Normal components of the magnetic flux densities are continuous such that
Here, the vectors n and t in these equations are respectively unit normal and tangential vectors at the boundary.
PROBLEMS 1. Use Maxwell's equations to specify the features of a plane wave propagating in a homogeneous nonconductive medium.
ANSWER Maxwell's equations are written as
Since the electric and magnetic fields of the plane wave depend not on the x and y coordinates but on the z coordinate, the derivatives with respect to the coordinates for directions other than the propagation direction are zero. That is, d/dx = d/dy = 0.
PROBLEMS
11
From Eqs. (PI.3) and (PI.6), we get
The remaining equations are
Equations (Pl.S)-(Pl.ll) are categorized into two sets:
The equations of set 1 can be reduced to
where k2 — co2£0/i0er = k^&r. And the equations of set 2 can be reduced to
Here, we discuss a plane wave propagating in the z direction. Considering that Eq. (PI. 14) implies that both the electric field component Ex and the magnetic field component Hy propagate with the wave number k, where it should be noted that the propagation constant ft is equal to the wave number k in this case and that the pure imaginary number j [— exp(^/7t)] corresponds to phase rotation by 90°, we can illustrate the propagation of the electric field component Ex and the magnetic field
12
FUNDAMENTAL EQUATIONS
FIGURE Pl.l. Propagation of an electromagnetic field.
component Hy, as shown in Fig. Pl.l. When we substitute Ex for Ey and —Hy for Hx, the equations of set 2 are equivalent to those of set 1. Since the field components in set 2 can be obtained by rotating the field components in set 1 by 90°, sets 1 and 2 are basically equivalent to each other. The features of the plane wave are summarized as follows: (1) the electric and magnetic fields are uniform in directions perpendicular to the propagation direction, that is, d/dx = d/dy = 0; (2) the fields have no component in the propagation direction, that is, Hz = Ez = 0; (3) the electric field and the magnetic field components are perpendicular to each other; and (4) the propagation direction is the direction in which a screw being turned to the right, as if the electric field component were being turned toward the magnetic field component, advances. 2. Under the assumption that the relative permeability in the medium is equal to 1 and that a plane wave propagates in the +z direction, prove that ^Hy =
The derivative with respect to the z coordinate can be reduced to d/dz = —jk = —j(D^/&ji^ by using Eq. (PI. 14). Thus, the relation follows fromEq. (PI.12). REFERENCE [1] R. E. Collin, Foundations for Microwave Engineering, McGraw-Hill, New York, 1966.
CHAPTER 2
ANALYTICAL METHODS
Before discussing the numerical methods in Chapters 3-7, we first describe analytical methods: a method for a three-layer slab optical waveguide, an effective index method, and Marcatili's method. For actual optical waveguides, the analytical methods are less accurate than the numerical methods, but they are easier to use and more transparent. In this chapter, we also discuss a cylindrical coordinate analysis of the step-index optical fiber.
2.1 METHOD FOR A THREE-LAYER SLAB OPTICAL WAVEGUIDE In this section, we discuss an analysis for a three-layer slab optical waveguide with a one-dimensional (ID) structure. The reader is referred to the literature for analyses of other multilayer structures [1, 2]. Figure 2.1 shows a three-layer slab optical waveguide with refractive indexes w l 5 n2, and «3. Its structure is uniform in the y and z directions. Regions 1 and 3 are cladding layers, and region 2 is a core layer that has a refractive index higher than that of the cladding layers. Since the tangential field components are connected at the interfaces between adjacent media, we can start with the Helmholtz equations (1.47) and (1.49), which are for uniform media. Furthermore, since the structure is 13
14
ANALYTICAL METHODS
FIGURE 2.1. Three-layer slab optical waveguide.
uniform in the y direction, we can assume d/dy — 0. Thus, the equation for the electric field E is
Similarly, we easily get the equation for the magnetic field H:
Next, we discuss the two modes that propagate in the three-layer slab optical waveguide: the transverse electric mode (TE mode) and the transverse magnetic mode (TM mode). For better understanding, we again derive the wave equation from Maxwell's equations
whose component representations are
2.1
METHOD FOR A THREE-LAYER SLAB OPTICAL WAVEGUIDE
15
As mentioned in Chapter 1, we assume here that the relative permeability jUr = 1. That is, \i — fj.r^Q — jU0. Since the structure is uniform in the propagation direction, the derivative with respect to the z coordinate, 3/3z, can be replaced by —//?. The effective index can be expressed as «eff = p/kQ, where &0 is the wave number in a vacuum.
2.1.1
TEMode
In the TE mode, the electric field is not in the longitudinal direction (Ez = 0) but in the transverse direction (Ey ^ 0). Since the structure is uniform in the y direction, d/dy = 0. Substitution of these relations into Eq. (2.10) results in dHy/dx = 0. Since this means that Hy is constant, we can assume that Hy = 0. Furthermore, substitution of Ez = Hy = 0 into Eq. (2.6) results in dEx/dz = 0, which means that Ex = 0. We thus get
Substituting Hx = —(ft/cofj.Q)Ey9 derived from Eq. (2.5), and Hz = (j/cono) dEy/dx, derived from Eq. (2.7), into Eq. (2.9), we get the following wave equation for the principal electric field component Ey:
where k0 = co^/e0/i0.
16
ANALYTICAL METHODS
Next, we derive the characteristic equation used to calculate the effective index « eff . The principal electric field component Ey in regions 1, 2, and 3 can be expressed as
Here, C ls C2, and C3 are unknown constants. Since the number of unknowns is 4 (neff, C1? C2, and C3), four equations are needed to determine the effective index «eff. To obtain the four equations, we impose boundary conditions on the tangential electric field component Ey and the tangential magnetic field component Hz at x = 0 and x = W. The tangential magnetic field component Hz is
which for the three regions is expressed as
Since the boundary condition requirement is that the tangential electric field components as well as the tangential magnetic field components are equal at the interfaces between adjacent media, the boundary conditions on these field components at jc = 0 are expressed as
2.1
METHOD FOR A THREE-LAYER SLAB OPTICAL WAVEGUIDE
17
and at x = W are expressed as
The resultant equations are
Thus, dividing Eq. (2.25) by Eq. (2.24), we get
On the other hand, dividing Eq. (2.27) by Eq. (2.26), we get
Substitution of a in Eq. (2.28) into Eq. (2.29) results in the following characteristic equation:
Or, using
18
ANALYTICAL METHODS
we can rewrite this equation as
2.1.2
TMMode
In the TM mode, the magnetic field component is not in the longitudinal direction (Hz = 0) but in the transverse direction (Hy ^ 0). Since the structure is uniform in the y direction, d/dy = 0. Thus, we get 3Ey/dx = 0 from Eq. (2.7). Since this means that Ey is constant, we can assume that Ey = 0. Furthermore, substitution ofHz = Ey = 0 into Eq. (2.9) results in dHx/Bz = 0, which means that Hx = 0. We thus get
Substituting Ex = (f$ / a)&Q&r}Hy, derived from Eq. (2.8), and Ez = —(j/cQSQS^dHy/dx, derived from Eq. (2.10), into Eq. (2.6), we get the following wave equation for the principal magnetic field component H,:
The field components on which the boundary conditions should be imposed are the principal magnetic field component Hy and the longitudinal electric field component Ez. The principal magnetic field component Hy can be expressed as
2.1
METHOD FOR A THREE-LAYER SLAB OPTICAL WAVEGUIDE
19
The tangential electric field component Ez is
which is expressed as
Imposing the boundary conditions on the tangential fields at x = 0 and x = W, we get
Dividing Eq. (2.43) by Eq. (2.42), we get
On the other hand, dividing Eq. (2.45) by Eq. (2.44), we get
20
ANALYTICAL METHODS
Substitution of the variable a in Eq. (2.46) into Eq. (2.47) results in the following characteristic equation:
Using Eq. (2.31), we also get
Comparing the characteristic equations (2.30) and (2.32) for the TE mode and Eqs. (2.48) and (2.49) for the TM mode, one discovers that the characteristic equations for the TM mode contain the ratio of the relative permittivities of adjacent media.
2.2
EFFECTIVE INDEX METHOD
Here, we discuss the effective index method, which allows us to analyze two-dimensional (2D) optical waveguide structures by simply repeating the slab optical waveguide analyses. Figure 2.2 shows an example of a 2D optical waveguide and illustrates the concept of the effective index method. We consider the scalar wave equation
where neff is the effective index to be obtained. We separate the wave function (x, y) into two functions:
2.2
Effective index n effl
EFFECTIVE INDEX METHOD
Effective index neff2
21
Effective index n eff3
FIGURE 2.2. Concept of the effective index method.
This corresponds to the assumption that there is no interaction between the variables x and y. Substituting Eq. (2.51) into Eq. (2.50) and dividing the resultant equation by the wave function 0(jc, y), we get
22
ANALYTICAL METHODS
Setting the sum of the second and third terms of Eq. (2.52) equal to k$N2(x), we get
This means that the sum of the first and fourth terms is equal to —klN2(x}\
Through these procedures we get the two independent equations
and
The effective index calculation procedure can be summarized as follows: (a) As shown in Fig. 2.2, replace the 2D optical waveguide with a combination of ID optical waveguides. (b) For each ID optical waveguide, calculate the effective index along the y axis. (c) Model an optical slab waveguide by placing the effective indexes calculated in step (b) along the x axis. (d) Obtain the effective index by solving the model obtained in step (c) along the x axis. It should be noted that, for the TE mode of the 2D optical waveguide, we first do the TE-mode analysis and then the TM-mode analysis. And, for
2.3
MARCATILI'S METHOD
23
the TM-mode analysis of the 2D optical waveguide, we do these analyses in the opposite order.
2.3
MARCATILI'S METHOD
Here, we discuss Marcatili's method for analyzing 2D optical waveguides [3]. Proposed in 1969, it is still in wide use. Figure 2.3 shows a cross-sectional view of a buried optical waveguide. The core has a refractive index « l 5 width 2a, and height 2b. It is surrounded by cladding that has a refractive index n2. In Marcatili's method, it is assumed that the electric fields and magnetic fields are confined to the core and do not exist in the four hatched regions shown in Fig. 2.3. Thus, the continuity conditions for the electric fields and the magnetic fields are imposed only at the interfaces of regions of 1 and 2, 1 and 3, 1 and 4, and 1 and 5. We discuss the E*pq mode, which has Ex and Hy as principal field components, and the Eypq mode, which has Ey and Hx as principal field components. Here,/? and q and are integers and respectively correspond to the numbers of peaks of optical power in the x and y directions. Thus, unlike ordinary mode orders, which begin from 0, they begin from 1.
FIGURE 2.3. Marcatili's method.
24
2.3.1
ANALYTICAL METHODS
Expq Mode
The electric field of the E*pq mode is assumed to be polarized in the x direction, which results in Ey = 0. Since the structure of the optical waveguide is assumed to be invariant in the z direction, the derivative with respect to 2 is replaced by —jfi. The component representations shown in Eqs. (2.5)-(2.10) are reduced to
Thus, we get
where
On the other hand, the component representation of the divergence equation
2.3
MARCATILI'S METHOD
25
is
Thus, we get the longitudinal electric field
Here, we assumed that l/e r • der/dx = 0. That is, we ignored the dependence of the relative permittivity er on the coordinate x in each region. This assumption will also be used for each element in the finite-element method discussed in Chapter 3. Eliminating Ez by substituting Eq. (2.68) into Eqs. (2.63) and (2.64), we get
The above results can be summarized as
26
ANALYTICAL METHODS
Substituting Eqs. (2.72) and (2.73) into Eq. (2.60), we get a wave equation for a principal field component Ex for the Expq mode such that
The electric field and the magnetic field components to be connected are Ex and Hz at the boundaries y — ±6 and are Ez and Hy at the boundaries x = ±a. Since the principal field component Ex is a solution (i.e., wave function) of the wave equation (2.75), we get the following field components in regions 1-5:
Substituting these wave functions into the wave equation (2.75), we get (after some mathematical manipulations) the following relations for the wave numbers:
Subtracting Eq. (2.81) from Eq. (2.83) and Eq. (2.81) from Eq. (2.82), we get
2.3
MARCATILI'S METHOD
27
The next step is to impose the boundary conditions specified by Eqs. (1.55) and (1.56) on the electric and magnetic fields. A. Connection at y = ±b: Ex and Hz Setting the electric field components Ex of regions 1 and 2 equal at y = b, from Eqs. (2.76) and (2.77), we get
And setting the magnetic field components Hz of regions 1 and 2, obtained by substituting Eqs. (2.76) and (2.77) into Eq. (2.73), equal at y = b, we get
Dividing Eq. (2.87) by Eq. (2.86), we get
Therefore
On the other hand, setting the electric field components Ex of regions 1 and 4 equal at y = —b, from Eqs. (2.76) and (2.79), we get
And setting the magnetic field components Hz of regions 1 and 4, obtained by substituting Eqs. (2.76) and (2.79) into Eq. (2.73), equal aty= -b, we get
28
ANALYTICAL METHODS
Dividing Eq. (2.90) by Eq. (2.89), we get
And adding Eq. (2.88) to Eq. (2.91), we get
B. Connection at x = ±a: Ez and Hy Setting the electric field components Ez of regions 1 and 3, obtained by substituting Eqs. (2.76) and (2.78) into Eq. (2.74), equal at x = a, we get
And setting the magnetic field components Hy of regions 1 and 3, obtained by substituting Eqs. (2.76) and (2.78) into Eq. (2.72), equal atx = a, we get
Dividing Eq. (2.93) by Eq. (2.94), we get
Substituting the relations
2.3
MARCATILI'S METHOD
29
which are obtained from Eqs. (2.81) and (2.83), into Eq. (2.95), we get
Therefore
On the other hand, setting the electric field components Ez of regions 1 and 5, obtained by substituting Eqs. (2.76) and (2.78) into Eq. (2.74), equal at x = —a, we get
And setting the magnetic field components Hy of regions 1 and 5, obtained by substituting Eqs. (2.76) and (2.78) into Eq. (2.72), equal at x = —a, we get
Substituting Eqs. (2.96) and (2.97) into the equation derived by dividing Eq. (2.99) by Eq. (2.100), we get
Therefore
30
ANALYTICAL METHODS
and
Now, adding Eq. (2.98) to Eq. (2.101), we get
where srl = n\ and sr2 = «f- O^ since &o w i,2 ^ ^y f°r most cases, we get
The propagation constant ft (or the effective index « eff ) can be obtained as follows: 1. Obtain kx by using a numerical technique such as a successive bisection method to solve Eq. (2.103) [or Eq. (2.102)]. (It should be noted that in each process of the successive bisection routine Eq. (2.84) can be used to obtain the yx corresponding to kx.) 2. Similarly, obtain ky by making use of Eqs. (2.85) and (2.92). 3. Finally, obtain the propagation constant ft from Eq. (2.81). Interesting points are as follows: Equation (2.92) corresponds to the characteristic equation of the TE mode for a three-layer slab optical waveguide parallel to the x axis. Equation (2.103), on the other hand, corresponds to the characteristic equation of the TM mode for a threelayer slab optical waveguide parallel to the y axis. The correspondence is the same as that described in the last part of Section 2.2. The x andy axes are related to each other through Eq. (2.81).
2.3
MARCATILI'S METHOD
31
2.3.2 Eypq Mode The magnetic field of the E^q mode is assumed to be polarized in the x direction, which results in Hy = 0. The component representations shown in Eqs. (2.5)-(2.10) are reduced to
On the other hand, the component representation of the magnetic divergence equation
is expressed as
32
ANALYTICAL METHODS
Thus, we get the longitudinal magnetic field component
Eliminating Hz by substituting Eq. (2.115) into Eqs. (2.110) and (2.111), we get
The above results can be summarized as
Substituting Eqs. (2.119) and (2.120) into Eq. (2.107), we get the following wave equation for a principal field component Hx for the Eypq mode:
The electric field and the magnetic field components to be connected are Hx and Ez at the boundaries y = ±b and are Hz and Ey at the boundaries x = ±a.
2.3
MARCATILI'S METHOD
33
Since the principal field component Hx is a solution (i.e., a wave function) of the wave equation (2.122), we get
Substituting these wave functions into the wave equation (2.122), we get (after some mathematical manipulations) for the wave numbers the same equations specified by Eqs. (2.81)-(2.85) for the E^q mode. Equations (2.96) and (2.97) also hold. The next step is to impose the boundary conditions on the tangential electric fields and the magnetic field components at the interfaces between different media. A. Connection at y = ±b: Ez and Hx Setting the electric field components Ez of regions 1 and 2, obtained by substituting Eqs. (2.123) and (2.124) into Eq. (2.120), equal at y = b, we get
And setting the electric fields Hx of regions 1 and 2 equal at y = b, from Eqs. (2.123) and (2.124), we get
Dividing Eq. (2.128) by Eq. (2.129), we get
34
ANALYTICAL METHODS
Therefore
On the other hand, setting the electric field components Ez of regions 1 and 4, which are obtained by substituting Eqs. (2.123) and (2.126) into Eq. (2.120), equal aty = —b, we get
Setting the magnetic field components Hx of regions 1 and 4 equal at y = -b, from Eqs. (2.123) and (2.126), we get
Dividing Eq. (2.131) by Eq. (2.132), we get
Therefore
Adding Eq. (2.130) to Eq. (2.133), we get
B. Connection at x = ±a: Ey and Hz Setting the electric field components Hz of regions 1 and 3, obtained by substituting Eqs. (2.123) and (2.125) into Eq. (2.119), equal at x = a, we get
2.3
MARCATILI'S METHOD
35
Setting the electric field components Hz of regions 1 and 3, obtained by substituting Eqs. (2.123) and (2.125) into Eq. (2.121), equal at x = a, we get
Dividing Eq. (2.136) by Eq. (2.135), we get
Here, making use of Eqs. (2.96) and (2.97), we get
Therefore
On the other hand, setting the electric field components Ey of regions 1 and 5, obtained by substituting Eqs. (2.123) and (2.127) into Eq. (2.119), equal at x = —a, we get
And setting the magnetic field components Hz of regions 1 and 5, obtained by substituting Eqs. (2.123) and (2.127) into Eq. (2.121), equal at* = -a, we get
Dividing Eq. (2.140) by Eq. (2.139), we get
36
ANALYTICAL METHODS
And substituting Eqs. (2.96) and (2.97) into this equation, we get
Therefore
Here, adding Eq. (2.138) to Eq. (2.141), we get
Or, since &o w i,2 ^ ky f°r most cases, we get
The propagation constant ft (or the effective index « eff ) is obtained in just the same way as for the E^q mode. Equation (2.134) corresponds to the characteristic equation of the TM mode for a three-layer slab optical waveguide parallel to the x axis. Equation (2.143), on the other hand, corresponds to the characteristic equation of the TE mode for a three-layer slab optical waveguide parallel to the y axis. The correspondence is the same as that described in the last part of Section 2.2.
2.4
METHOD FOR AN OPTICAL FIBER
In this section, we discuss an analysis of a step-index optical fiber. An optical fiber consists of a core and a cladding and is axially symmetric. Since the refractive index of the core is slightly higher than that of the cladding, the optical field is largely confined to the core. A single-mode
2.4
METHOD FOR AN OPTICAL FIBER
37
FIGURE 2.4. Step-index optical fiber.
fiber, which has only one guided mode, plays a key role in telecommunications systems. Figure 2.4 shows a cross-sectional view of a step-index optical fiber. The core of a radius a has a uniform refractive index nl, slightly higher than the refractive index of the cladding, n2. Thus, the relative permittivities of the core and cladding are respectively srl = n\ and er2 = n\. The exact vectorial wave equations for the electric field and the magnetic field were shown in Chapter 1 as Eqs. (1.30) and (1.36). Since the structure of the optical fiber is uniform in the propagation direction, we can, as shown in Eq. (1.39), substitute —jfi for the derivatives of the electric and magnetic fields with respect to z. Thus, we can write
where E and H are respectively the electric fields and the magnetic fields and «eff is the effective index to be obtained. Although the structure shown in Fig. 2.4 can be analyzed exactly by using the hybrid-mode analysis [4], the analysis procedure is a little complicated. Fortunately, however, the weakly guiding approximation can be used because the refractive index difference between the core and the
38
ANALYTICAL METHODS
cladding is very small, about 1%. This approximation simplifies the analysis significantly, and the modes obtained are called linearly polarized modes [5].
2.4.1
Linearly Polarized Modes (LP Modes)
A. Field Expressions Since the derivative of the relative permittivity sr is small, neglecting the derivatives in the vectorial wave equations (2.144) and (2.145) gives a good approximation. Using this approximation, we can reduce the vectorial wave equations to the scalar Helmholtz equations for the tangential electric fields and the magnetic fields:
Since the optical fiber is axially symmetric, the Laplacian V2 is rewritten for a cylindrical coordinate system as follows:
To solve Eq. (2.146), we assume that the tangential electric field component Ej_ (i.e., Ex or Ey) is given by
where the subscript i = :c, y. Substituting the electric field component (2.148) into Eq. (2.146) and dividing the resultant equation by R(r)®(0\ we get
2.4
METHOD FOR AN OPTICAL FIBER
39
Therefore
Since the left-hand side of Eq. (2.149) is a function of only the variable r and the right-hand side is a function of only the variable 6, both sides have to be constant. Thus, we get
and
Equations (2.149) and (2.151) are summarized as
and
The solution of Eq. (2.153) is an oscillation with a single frequency and is expressed as
where / and are respectively an integer and an arbitrary constant phase. The next step is to solve Eq. (2.152). Using the variable transformations
and
40
ANALYTICAL METHODS
we write the first and second derivatives with respect to r as
Equation (2.152) is then rewritten as
Therefore
Solutions for Eq. (2.158) are /th-order Bessel functions and are written as
Here, J^ur/d) and N{(ur/a) are the /th-order Bessel functions of the first and second kinds, and K^wr/a) and I^wr/a) are the /th-order modified Bessel functions of the first and second kinds. The parameters u2 and w2 are defined as
Thus, we get the following very important relation between u and w.
where
2.4
METHOD FOR AN OPTICAL FIBER
41
is the normalized frequency. The parameters u and w are respectively considered to be the normalized lateral propagation constant in the core and the normalized lateral decay constant in the cladding. The effective index neff has to satisfy the relation
Since the Bessel function of the second kind, Ni(ur/d), diverges at r = 0 and the modified Bessel function of the second kind, Ij(wr/a), diverges at r — oo, the coefficients B and D of those functions have to be zero. Thus, from Eq. (2.159), we get
B. Characteristic Equation The boundary conditions to be satisfied by the radial wave function R(r) are
and
From Eqs. (2.166) and (2.167), we get
Equations (2.166) and (2.167) can be rewritten as the matrix equation
42
ANALYTICAL METHODS
When the coefficients A and C are nontrivial, the determinant of their coefficient matrix has to be zero such that
Equation (2.171) can be rewritten as
where the prime denotes the derivative with respect to r. Thus, we get the well-known characteristic equation
The effective index «eff can be obtained by solving a combination of Eqs. (2.162) and (2.172). In other words, since u and w are functions of the effective index n eff , the characteristic equation (2.172) is a function of « eff . Solutions of this equation are called linearly polarized modes (LP modes).
1. Explicit Forms of the Characteristic Equation LP0m MODES (/ = o AND m > l)
Equation (2.172) is rewritten for / = 0 as
Making use of the Bessel function formulas
we rewrite Eq. (2.173) as
2.4
METHOD FOR AN OPTICAL FIBER
43
Therefore
LP,m MODES (/ > 1 AND m > 1) Here, we consider the LPlm mode. Equation (2.172) is rewritten for / = 1 as
Making use of the Bessel function formulas
we rewrite Eq. (2.177) as
Therefore
In general, making use of the Bessel function formulas
one can rewrite Eq. (2.177) as
44
ANALYTICAL METHODS
Therefore
2. Value Range of u The characteristic equations for LP/OT modes have solutions only within limited ranges of the parameter u, and we need to know what these ranges are. These ranges can be determined by investigating the limits w —>• 0 and w -> +00, where the former corresponds to u -> v . Through this process, the single-mode condition and cutoff conditions for the higher order LP/7W modes will also be clarified. LP0m MODES (7 = 0 AND m > l) First, we investigate the limit w —> 0 (i.e., u —> v). Since the zeroth-order and /th-order modified Bessel functions of the first kind can, for the limit of z —>• 0, be respectively expressed asymptotically as
The right-hand side of Eq. (2.176) can be rewritten as
The left-hand side of Eq. (2.176) also has to go to +00. That is,
The possible solutions for Eq. (2.186) are v —»• 0 and J\(v) —> 0. Since we get JQ(V) —»• 1 and J\(v) -> +0 for v —»• 0, Eq. (2.186) holds. This implies that the cutoff value vc for the normalized frequency v is zero. In other words, the LP01 mode has no cutoff condition. Next, we discuss the cutoff conditions for higher order modes: LP0m modes where m > 2. Here, we assume thatj! m_i is the (m — l)th zero of the Bessel function of the first kind. That is, J\(j\tTn-\) = 0. Since the signs ofJ0(v) andJ^v} are the
2.4
METHOD FOR AN OPTICAL FIBER
45
same for the limit v -+j\tTn-\ +0, Eq. (2.186) holds. This means the cutoff value vc of the LP0w2 mode is given by
Thus, we can summarize the cutoff conditions for LP0wj modes as follows:
On the other hand, when w ;$> 1, the asymptotic expansion of the /thorder modified Bessel function of the first kind is
Since for w ^> 1 the right-hand side of Eq. (2.176) can be rewritten as
the left-hand side of Eq. (2.176) also has to go to zero. That is,
This implies that the asymptotic value of u is given by
46
ANALYTICAL METHODS
Thus, we can summarize the asymptotic values of u for LP0/n modes as follows:
LP/m MODES (/ > l AND m > 1) As in the above discussion, we first investigate the limit w -» 0 (i.e., u -> t;). For the limit z -» 0, the /thorder modified Bessel functions of the first kind were shown in Eqs. (2.185) and (2.185). The (/ - l)th-order modified Bessel function of the first kind for the limit z —* 0 is
Making use of this approximation, we express the right-hand side of Eq. (2.183) as
The left-hand side of Eq. (2.183) also has go to —oo. That is,
The possible solutions of Eq. (2.196) are v —> 0 and Ji_i(v) -> 0. Since the left-hand side of Eq. (2.183) diverges to +00 for the limit v —> 0, the limit v -» 0 cannot be a solution. On the other hand, since the signs of Ji(v) and J^i(v) are opposite for v ->y/_i, OT + ®>Ji-\,m can be a solution for Eq. (2.196). Here, ji_\tm is the zero of the (/- l)th-order Bessel function of the first kind, //^(i/).
2.4
METHOD FOR AN OPTICAL FIBER
47
The cutoff condition for LP/m modes is
On the other hand, since, according to the asymptotic expression (2.189), the right-hand side of Eq. (2.183) is 0 for w » 1, the left-hand side of Eq. (2.176) also has to go to zero. That is,
This implies that the asymptotic value of u is given by
We can thus summarize the possible value ranges of the parameter u as
It should be noted that since the minimum value of the zeros of the Bessel functions is y'0>1 (2.404826), the single-mode condition for an optical fiber with a step index is given by the cutoff value for the LPn mode:
Figure 2.5 shows examples of u—co curves. The crossing points of the curve u2 + w2 = v2 and the other curves give the effective indexes and confirm the value ranges of u specified above. Figure 2.6 shows examples of calculated field distributions. It should be noted that / and m of LP/W respectively correspond to the number of dark lines in the azimuthal direction and the number of bright peaks in the radial direction. 2.4.2
Hybrid-Mode Analysis
This section discusses a more exact analysis for the step-index optical fiber. Since the above LP-mode analysis can meet the ordinary demands for fiber analyses and the hybrid-mode analyses are very specialized, some readers may wish to skip this section.
48
ANALYTICAL METHODS
FIGURE 2.5. Relation between u and w (v — 4).
FIGURE 2.6. Field distributions for the LP mode (v = 20): (a) LP0 t ; (b) LPj ^ (c) LP2,2; (d) LP5,2.
4. Field Expressions The cylindrical electric field E and the cylindrical magnetic field H are expressed as
2.4
METHOD FOR AN OPTICAL FIBER
49
where
Here, r, 6, and z are respectively unit vectors in the radial, azimuthal, and longitudinal directions. Applying the rotation formula
for a vector A = Arr + A0Q + Azz to the Maxwell equations
we get
50
ANALYTICAL METHODS
Expressing the tangential field components (Er, Ee, Hr, and He) as functions of the longitudinal field components (Ez and Hz\ we get
The radial and azimuthal field components are obtained as follows: Equation (2.216) x ft + Eq. (2.217) x co/^0:
Equation (2.215) x ft + Eq. (2.218) x c^0:
Equation (2.215) x coe + Eq. (2.218) x /?:
Equation (2.216) x cos + Eq. (2.217) x p:
Substituting the Hr of Eq. (2.221) and the He of Eq. (2.222) into Eq. (2.214), we get
2.4
METHOD FOR AN OPTICAL FIBER
51
And substituting the Er of Eq. (2.219) and the Ee of Eq. (2.220) into Eq. (2.211), we get
To solve Eqs. (2.223), we assume that the longitudinal field components Ez and Hz are given by
Thus, we get the following wave equations for Rz(r) and ®z(0):
and
The solution of Eq. (2.227) is an oscillation with a single frequency and is expressed as
where n and 0 are respectively an integer and an arbitrary constant phase Through the procedure shown for the LP mode, we obtain the radial wave function Rz(r) in the core as the Bessel function of the first kind Jn(ur/a), and obtain the radial wave function in the cladding as the
52
ANALYTICAL METHODS
modified Bessel function of the first kind, Kn(ur/d). Thus, we finally get the following field components:
Substituting Eqs. (2.229) and (2.230) into Eqs. (2.219M2.222), we get the wave functions as follows: 1. In the core (r < a):
2.4
METHOD FOR AN OPTICAL FIBER
53
2. In the cladding (r > a):
The normalized lateral propagation constant u in the core and the normalized lateral decay constant w in the cladding were respectively denned in Eqs. (2.160) and (2.161).
B. Characteristic Equation The boundary conditions to be satisfied are that each of the tangential field components (Ez, E0, Hz, and He) is continuous at r = a. They are expressed as
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ANALYTICAL METHODS
Substituting the equations for the field components [i.e., Eqs. (2.231)(2.242)], into the equations for the boundary conditions [i.e., Eqs. (2.243)-(2.246)], we get the matrix equation
When the coefficients A, B, C, and D have nontrivial solutions, the determinant of the coefficients of Eq. (2.247) has to be zero. After some mathematical manipulations, we get the characteristic equation
This is the characteristic equation for the hybrid mode (i.e., Ez ^ 0 and Hz 7^ 0). Since u and w are functions of the effective index « eff , this characteristic equation is also a function of n eff . For n = 0, Eq. (2.248) is reduced to
or
It can be shown that Eqs. (2.249) and (2.250) respectively correspond to the TE mode (Ez = 0) and the TM mode (Hz = 0).
PROBLEMS
55
PROBLEMS 1. Derive the expressions for the power confinement factors (F factors) in the core for the TE mode and the TM mode of the three-layer slab optical waveguide shown in Fig. 2.1.
ANSWER a. TE mode. The principal electric field component is Ey and the other field components are
Since the complex Poynting vector S is defined as
the power propagating in the z direction, Sz, is given by
The power confinement factor in the core is thus
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ANALYTICAL METHODS
b. TM mode. The principal electric field component is Hy is and the other field components are
The power propagating in the z direction, Sz, is given by
The power confinement factor in the core is thus
2. The characteristic equations for the LP0m mode, Eq. (2.176), and for the LPlm mode, Eq. (2.183), were derived separately. Show that, when / = 0, Eq. (2.176) is included in Eq. (2.183).
ANSWER Substituting / = into Eq. (2.183), we get
REFERENCES
57
Since the Bessel functions here have the formulas
assuming n = 1, we get
The characteristic equation (2.176) can be derived by substituting Eqs. (P2.15) and (P2.16) into Eq. (P2.12).
REFERENCES [1] T. Tanaka and Y. Suematsu, "An exact analysis of cylindrical fiber with index distribution by matrix method and its application to focusing fiber," Trans. lECEJpn., vol. E-59, no. 11, pp. 1-8, 1976. [2] K. Kawano, Introduction and Application of Optical Coupling Systems to Optical Devices, 2nd ed., Gendai Kohgakusha, Tokyo, 1998 (in Japanese). [3] E. A. Marcatili, "Dielectric rectangular waveguide and directional coupler for integrated optics," Bell Syst. Tech. J., vol. 48, pp. 2071-2102, 1969. [4] E. Snitzer, "Cylindrical dielectric waveguide modes," J. Opt. Soc. Am., vol. 51, pp. 491^98, 1961. [5] D. Gloge, "Weakly guiding fibers," Appl. Opt., vol. 10, pp. 2252-2258, 1 r\T 1
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CHAPTER 3
FINITE-ELEMENT METHODS
Like the finite-difference methods (FDMs) that will be discussed in Chapter 4, the finite-element methods (FEMs) [1-3] are widely utilized numerical methods. The scalar (SC) FEM has several advantages over the fully vectorial (V) FEM. The main ones are that the SC-FEM has no spurious problem and the matrixes in the eigenvalue equation are small and symmetrical These contribute to numerical efficiency. Here, we discuss the use of the SC-FEM in the 2D cross-sectional analysis of optical waveguides.
3.1
VARIATIONAL METHOD
As shown in Fig. 3.1, there are two kinds of ways that can be used to solve optical waveguide problems [2-4]: the variational method and weighted residual method, of which the Galerkin method is representative. Both the variational and the weighted residual methods eventually require that the same matrix eigenvalue equations be solved. This section will focus on a variational method; the next section will discuss a weighted residual method. In the variational method, the wave equation is not directly solved. Instead, the analysis region is divided into many segments and the variational principle is applied to the sum of the discretized functional for all segments. 59
60
FINITE-ELEMENT METHODS
FIGURE 3.1. Analysis based on the finite-element method.
Figure 3.2 shows an analysis region Q surrounded by a boundary P. Here, n is the outward-directed unit vector normal to the surface of the analysis region Q. A variational method for obtaining the effective index neff is first discussed here by using the scalar wave equation [2,3]
Multiplying this equation by variation dcf) of the function $ and integrating the product over the whole analysis region Q, we get
Here, applying the partial integration with respect to variables x and y, we get
FIGURE 3.2. Analysis region.
3.1
VARIATIONAL METHOD
61
where the terms inside the brackets are those whose integration orders with respect to x and y were reduced from 2 to 1 because of the partial integration. Summarizing the second and the third terms, we get
In addition, substituting the relations
into the above equation, we get
Here, we introduce the following function 7:
62
FINITE-ELEMENT METHODS
Since the line integral calculus term can be rewritten as
7 can be rewritten as
where d/dn is the derivative with respect to the normal vector n. Here, / is a function of , which is a function of* and y. A function of a function is generally called a functional. Using the functional /, we can rewrite Eq. (3.3) as
which means that the stationary condition is imposed on the functional /. We can instead first define the functional / given by Eq. (3.4) or (3.6). Then, imposing the stationary condition on the functional /, we get
When Eq. (3.8) holds for an arbitrary variation S(f> of the function , the wave equation (3.1) has to hold. Thus, the wave equation (3.1) is obtained by imposing the stationary condition on the functional /. This means that solving the wave equation (3.1) is equivalent to setting the variation 61 of the functional I, which can be obtained by Eq. (3.4) or (3.6), to zero, that is, to imposing the stationary condition on the functional I. This is called the variational principle, and a method for solving problems by using the variational principle is called a variational method. In the Rayleigh-Ritz method, the unknown function is formed by a linear combination of known basis functions that satisfy the boundary conditions, and the variational principle is applied to the functional.
3.1
VARIATIONAL METHOD
63
The calculation procedure for the application of the variational method to the FEMs is summarized as follows: The analysis region is first divided into segments, which are called elements, and the functional Ie is calculated for each element e. Then, the total functional / for the whole analysis region is obtained by summing up the functional Ie for all elements:
The final eigenvalue matrix equation is obtained by imposing the stationary condition on the functional /. It should be noted that the difference between an ordinary variational method and an FEM is that the former treats the analysis region as one area and the latter treats the region as the sum of elements. Since the total functional is a linear combination of the functionals for the elements, the variation 61 of the whole system is a sum of variation dle of each element e. Thus, Eq. (3.7) can be rewritten as
The functional Ie of an element e surrounded by the boundary Ye is given by
Next, the wave function
e is expanded as
by using the basis function [Ne] in element e, where Me is the number of nodes in element e and T is the transposing operator for a matrix. Then the basis function [Ne] and the expansion coefficient {4>e} are expressed as
64
FINITE-ELEMENT METHODS
In the FEMs, a basis function Nt is called a shape function or an interpolation function. As discussed later, the expansion coefficient 0,corresponds to a field component at each node. Two-dimensional cross-sectional analyses of optical waveguides often use first-order triangular elements having three nodes or second-order triangular elements having six nodes. We make use of the equation
where d(j)e/dx is a scalar quantity and is rewritten as (d(j)e/dx) we can rewrite
T
. Similarly,
and
where we made use of the fact that (f)e is also a scalar quantity. The functional Ie given by Eq. (3.11) can thus be rewritten as
3.1
VARIATIONAL METHOD
65
Here, matrixes [Ae] and [Be] and the quantity A2 are given by
and
Since the functional Ie given by Eq. (3.18) is obtained for only element e, we have to sum up all the elements to obtain the functional 7 for the whole analysis region. According to Eq. (3.9),
where
Now, consider the second term of Eq. (3.22), which is the line integral calculus term and can be expressed as follows:
Here, we assume that the wave function (j>e and its derivative with respect to the normal to the surface of element e, d
66
FINITE-ELEMENT METHODS
the line integral calculus term of the periphery of the whole analysis region remains, which will be discussed in Section 3.4. Thus, the second term of Eq. (3.22) can be reduced to
Finally, the functional 7 for the whole analysis region given by Eq. (3.22) is obtained as
Next, we impose on the functional 7 at the boundaries the Dirichlet condition
or the Neumann condition
Using these boundary conditions, we can reduce the functional 7 to the simple form
Now that we have obtained the functional 7, the next step is to impose the variational principle on it. Although having a total of only two nodes is impossible for 2D cross-sectional optical waveguides, this case will be used here for ease of understanding. The wave function vector {<£} and matrixes [S], [A], and [B] for the eigenvalue /I2 are expressed as
3.1
VARIATIONAL METHOD
67
and
Here, we make use of the symmetry for [S]. [The symmetry of the matrix, [S] = [S]T, will be discussed later.] Thus, we get
The derivatives of Eq. (3.34) with respect to 4>l and 2 are
And the matrix expression for these derivatives is
or
68
FINITE-ELEMENT METHODS
This is the same equation we get for general cases. Since solving the wave equation by using the FEM with the variational principle is the imposition of the stationary condition on the functional / as
it is equivalent to solving the eigenvalue matrix equation
3.2
GALERKIN METHOD
Since the functional is used to solve the problems, we have to find it when we use the FEM with the variational principle. On the other hand, the partial differential equations governing almost all the physical phenomena that we encounter are already known. This is certainly true with regard to the wave equations for the electromagnetic fields and for the Schrodinger equation, solutions of which are the targets of this book. The weighted residual methods, especially the Galerkin method, is quite powerful for solving them. The Galerkin method is widely used not only in the FEM but also in methods of microwave analyses, such as the spectral domain approach (SDA) [5, 6]. Since the wave function (j> is a true solution for the wave equation (3.1), the right-hand term of Eq. (3.1) is definitely zero. The true wave function (/>, however, cannot actually be known; we can obtain only an approximate wave function <^>. When the true wave function 0 in Eq. (3.1) is replaced by the approximate one, the right-hand term does not become zero but generates the error R, which is called the error residual:
It is natural to think that the difference between (f) and 0 can be decreased by averagely setting the error residual R equal to zero in the whole analysis region. As is well known, the electromagnetic fields concentrate mostly in the core where the refractive index is higher than in the cladding. Thus, some weighting should be used when setting R equal to zero. Introducing the weight function if/, we get
3.2 GALERKIN METHOD
69
Rewriting the error residual R explicitly, we get
The procedure discussed above is the weighted residual method. Partially integrating Eq. (3.43) with respect to x andy, we get
which can be rewritten as
Here, the relation given by Eq. (3.5) was used, and Jr dT and d/dn are respectively the line integral calculus at the boundary F and the derivative with respect to the normal vector n. The rank of the integral calculus of the first term inside the square brackets in Eq. (3.45) is decreased by 1 as a result of the partial integration. The rank of the derivatives of the second term is also decreased from 2 to 1, and these derivatives are called the weak forms. The weighted residual method, in which both the approximate wave function 0 and the weight function \l/ are expanded by the same basis functions, is called the Galerkin method. In using the FEM, we first divide the analysis region into many elements, then apply the Galerkin method to each element, then sum up the contributions of all the elements. The expansion coefficients obtained as an eigenvector correspond to the fields at nodes in the analysis region.
70
FINITE-ELEMENT METHODS
Since the calculation procedure will be discussed in detail later, here we simply summarize it. The equation for element e in the divided elements is expressed from Eq. (3.45) as
Here, it should be noted that $e and \j/e in element e are expanded by using the same basis functions:
where Me is the number of nodes in e. The [Ay and {(f)e} were defined in Eqs. (3.13) and (3.14). The basis function Nf and the expansion coefficient
for element e. Applying the definitions for [Ae], [Be], and A2 shown in Eqs. (3.19) to (3.21), we can rewrite the above equation as
or
3.2
GALERKIN METHOD
71
Since Eq. (3.50) gives the contribution of only element e, it is necessary to sum up the contributions of all the elements in the analysis region. Thus, we get
or
where the definitions
in Eqs. (3.23)-(3.25) were used. With respect to the second term in Eq. (3.51), we assume that the wave function (f)e and its normal derivative d(f)e/dn are continuous at the boundaries between elements. This assumption cancels out the line integral calculus terms, so the second term in Ea. (3.51) can be reduced to
Substituting Eq. (3.53) into (3.51), we get
When the Dirichlet condition or the Neumann condition is the boundary condition, the second term in Eq. (3.51) becomes zero and Eq. (3.54) is simplified to
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FINITE-ELEMENT METHODS
We finally get the following eigenvalue matrix equation to be solved:
Comparing Eq. (3.56) with Eq. (3.40), one can see that the eigenvalue matrix equation for the Galerkin method is identical to that for a variational method. 3.3
AREA COORDINATES AND TRIANGULAR ELEMENTS
We have roughly discussed the calculation procedures for the variational method and Galerkin method. Before describing the detailed formulations of the global matrixes for the eigenvalue matrix equations, we have to investigate the elements, which are indispensable when we divide the analysis region into segments. In the analysis of 2D cross-sectional structures, triangular elements using polynomials are generally used to approximate field distributions. The concept of polynomial approximations is illustrated in Fig. 3.3. In Fig. 3.3a, the true wave function is approximated by a linear function, and, in Fig. 3.36, it is approximated by a quadratic function. Although the higher order polynomial approximations can bring about more accurate results, they result in a larger number of nodes at which optical fields are defined. In addition, when there are more nodes, we need to use a more complicated mathematical analysis and more computer memory. Discussions in this book are therefore limited to two widely utilized triangular elements: the first-order triangular element, which requires three nodes, and the second-order triangular element, which requires six nodes.
FIGURE 3.3. Approximations by polynomial functions: (a) linear function; (b)
quadratic function.
3.3 AREA COORDINATES AND TRIANGULAR ELEMENTS
73
FIGURE 3.4. First-order triangular element.
3.3.1
First-Order Triangular Elements
Figure 3.4 shows a first-order triangular element. In this element, the wave function $(x,y) at an arbitrary coordinate (x,y) inside the element is expanded using shape functions NI, N2, and N3 (with fields 3) at the vertexes on which nodes are placed:
where [N] = [N\ N2 N3]T and {(/>} = (^l 2 <$>z)T• The shape functions [N] and the field vectors {<^>} respectively correspond to the basis functions and the expansion coefficients. The coordinates of nodes 1, 2, and 3 are respectively (x^y^, (jc2,y2) and fe^s)In determining the explicit forms of shape functions N^,N2, and N3, the use of area coordinates is convenient. Figure 3.5 shows a triangle we use here for discussing the area coordinates. An arbitrary coordinate in this triangle is denoted by p(x,y). Figure 3.6 shows another triangle formed by nodes denoted 1, 2, and 3. As is well known, the area Sm of the triangle is given by
When we assume that i, j, and k are unit vectors in the x, y, and z directions, the vector A from node 1 to node 2 and the vector B from node 1 to node 3 are expressed as
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FINITE-ELEMENT METHODS
FIGURE 3.5. Area coordinates.
Using these expressions, we get
Thus, we can obtain the area 5123 of the triangle 123 as
FIGURE 3.6. Area of a triangle.
3.3 AREA COORDINATES AND TRIANGULAR ELEMENTS
75
We similarly get area S23p of triangle 23p formed by nodes 2 and 3 and point/?; area Slp3 of triangle l/?3 formed by node 1, point/?, and node 3; and area S}2p of triangle Yip formed by nodes 1 and 2 and point/?:
Area coordinate L, can be defined as the ratio of the area of the triangle formed by point p and the side opposite node i to the whole area of the triangle. In Fig. 3.5, the area coordinate L l 5 which is related to node 1, is defined as the ratio of the area of triangle 23/? to the area of triangle 123. Similarly, the area coordinate Z2, related to node 2, is defined as the ratio of the area of triangle l/?3 to the area of triangle 123. And area coordinate L3, related to node 3, is defined as the ratio of the area of triangle Yip to the area of triangle 123. The explicit expressions for L b L2, and L3 are
Generally, since the value of the determinant for a matrix is invariant under transposition, we get the relations
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FINITE-ELEMENT METHODS
We similarly get the following relations for the other determinants:
Rewriting Eqs. (3.65)-(3.67) by using these relations, we get
As shown in these equations, area coordinates Ll, L2, and L3 are linear functions of x and 7. According to Cramer's formula, Eqs. (3.69)-(3.71) mean that Ll, L2, and L3 are roots of the following algebraic matrix equations:
where it can be found from the bottom row that the sum o^L^,L2, and L3 is 1.
3.3 AREA COORDINATES AND TRIANGULAR ELEMENTS
77
Now, we return our attention to the shape functions in Eq. (3.57). Since here we are interested in a first-order triangular element, we approximate shape functions Nlt N2, and 7V3 by the following first-order plane functions:
for which the matrix expression is
The conditions to be imposed on shape function NI are the following:
The matrix expression for Eqs. (3.77)-(3.79) is
Imposing the same conditions on N2 and 7V3, we get
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FINITE-ELEMENT METHODS
Equations (3.80)-(3.82) can be rewritten as
Using the definitions
we can rewrite Eq. (3.83) as
Taking the transposition T of Eq. (3.85), we get
Thus, matrix [X] is found to be an inverse matrix of [A] . Multiplying Eq. (3.76) by matrix [X]T, we get
This results in
Comparing Eq. (3.72) with Eq. (3.87), we find the following important relations between shape functions Nl, N2, and 7V3 and area coordinates L l 5 L2, and L3:
3.3 AREA COORDINATES AND TRIANGULAR ELEMENTS
79
FIGURE 3.7. Shape function N} for the first-order triangular element. Figure 3.7 shows A^ as an example. As shown in this figure, A^ is a plane function whose value is 1 at node 1. Similarly, N2 and N3 are respectively plane functions whose values are 1 at nodes 2 and 3. 3.3.2
Second-Order Triangular Elements
Figure 3.8 shows a second-order triangular element. As in the first-order triangular element shown in Fig. 3.4, nodes 1, 2, and 3 are positioned at the vertexes of the triangle. The second-order triangular element has three additional nodes set midway between the pairs of vertexes. The wave function cj)(x, y) at an arbitrary coordinate (jc, y) is expanded using six shape functions and the values of wave functions ( f > l , . . . , cj>6 at the nodes as
FIGURE 3.8. Second-order triangular element.
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FINITE-ELEMENT METHODS
where [N] = [Nl N2 N3 N4 N5 N6]T and {0} = (0! <£2 3 (f)4 05 6)r. The coordinates of nodes 1-6 are respectively (xl,yl),...,(x6,y6). As mentioned before, the first-order shape function is a linear function of x and y. The second-order shape functions should be quadratic polynomial functions of x and y. The area coordinates Ll} L2, and L3 are also linear functions of x and y, so the second-order shape functions must be quadratic functions of the area coordinates. A. Shape Function NI Shape function A^ is 1 at node 1 and is 0 at all other nodes. Since Ll = 1 and L2 = L^ — 0 at node 1, A^ should be a quadratic function of only area coordinate Ll. Thus,
The following relations should hold:
From Eqs. (3.91)-(3.93), we obtain
Thus, we get
B. Shape Function N2 Shape function N2 is 1 at node 2 and is 0 at all other nodes. Since L2 = 1 and Ll = L3 = 0 at node 2, N2 should be a quadratic function of only area coordinate L2. Thus,
3.3
AREA COORDINATES AND TRIANGULAR ELEMENTS
81
The following relations should hold:
From Eqs. (3.97)-(3.99), we obtain
Thus, we get
C. Shape Function N3 Shape function N3 is 1 at node 3 and is 0 at all other nodes. Since L3 = 1 and Lv = L2 = 0 at node 3, N3 should be a quadratic function of only area coordinate Z,3. Thus,
The following relations should hold:
From Eqs. (3.103)-(3.105), we obtain
Thus, we get
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FINITE-ELEMENT METHODS
D. Shape Function N4 Shape function N4 is 1 at node 4 and is 0 at all other nodes. Since L{ = L2 = 0.5 and L3 = 0 at node 4, N4 should be a quadratic function of area coordinates Ll and L2. Thus,
The following relations should hold:
From Eqs. (3.109H3.H4), we obtain
Thus, we get
E. Shape Function N5 Shape function N5 is 1 at node 5 and is 0 at all other nodes. Since L{ = 0 and L2 = £3 =0.5 at node 5, A^ should be a quadratic function of area coordinates L2 and L3. Thus,
The following relations should hold:
From Eqs. (3.118H3-123), we obtain
3.3 AREA COORDINATES AND TRIANGULAR ELEMENTS
83
Thus, we get
F. Shape Function N6 The shape function N6 is 1 at node 6 and is 0 at all other nodes. Since L: — L3 = 0.5 and L2 = 0 at node 6, N6 should be a quadratic function of the area coordinates L{ and L3. Thus,
The following relations should hold:
From Eqs. (3.127)-(3.132), we obtain
Thus, we get
The above results can be summarized as
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FINITE-ELEMENT METHODS
3.4
DERIVATION OF EIGENVALUE MATRIX EQUATIONS
Next, we move to the derivation of eigenvalue matrix equations for the Expq and Epq modes. As shown in Eqs. (2.75) and (2.76) in Marcatili's method, the wave equations are
Taking into consideration the continuity conditions at neighboring elements, which will be discussed later, we can rewrite the wave equation for the Epq mode as
Using Eqs. (3.141) and (3.142), we get the scalar wave equations [2, 3]
where for the E^q mode
and for the Epq mode
To derive the eigenvalue matrix equation for these modes, we use the Galerkin method discussed in Section 3.2. After the analysis region is divided into many elements, the wave function (f>e at the nodes in e is
3.4
DERIVATION OF EIGENVALUE MATRIX EQUATIONS
85
expressed by shape functions NI and wave functions
where Me is the number of nodes in e and T is a transposing operator for a matrix. We also used the following definitions:
where the numbers of the nodes Me in the first-order and second-order triangular elements are respectively 3 and 6. Substituting Eq. (3.147) into the wave equation (3.144), we get
Multiplying the left-hand side of this equation by the shape function [Ne] and integrating it in element e, we get
Partially integrating the first term of Eq. (3.150) with respect to x and y, we get
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FINITE-ELEMENT METHODS
Making use of Eq. (3.5), we get
where d/dn is the derivative with respect to the outside normal and fr dY is the line integration at boundary Te. Since the method we are discussing is an FEM, it is necessary to sum up the contributions from all the elements:
Here, we focus on the first term of the above equation. As mentioned before, Jr dT is a line integration at the boundary of element e. To simplify tne argument here, we assume that the left-hand side of Eq. (3.152) is multiplied by the wave function vector {(f)e}T. Thus, we get the relation
3.4
DERIVATION OF EIGENVALUE MATRIX EQUATIONS
87
This means that the first term in Eq. (3.152) can be rewritten as
Here, we assume that the wave function (j)e and its normal derivative rjl d(j)e/dn with constant r\2e are continuous at the boundaries with neighboring elements. Through this assumption, the line integral calculus terms inside the analysis region are canceled out, since, as shown in Fig. 3.9, the directions of the line integral calculus terms are opposite for each pair of neighboring elements. As a result, only the line integral calculus term of the periphery of the whole analysis region remains. Although this assumption is one of the limitations of the SC-FEM, it is a relatively good approximation. Thus, the first term in Eq. (3.152) can be rewritten as
FIGURE 3.9. Canceling out of line integral calculus terms.
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FINITE-ELEMENT METHODS
Substituting the line integral calculus term (3.153) into Eq. (3.152), we can reduce Eq. (3.152) to
When we impose the Dirichlet or Neumann conditions on fields or their derivatives given by Eqs. (3.28) and (3.29), we can neglect the last term on the left-hand side of Eq. (3.154). Then Eq. (3.154) can be simplified to the eigenvalue matrix equation
where the square of the propagation constant ft is an eigenvalue and [4>] is an eigenvector. Here, we used the definitions
Since the relative permittivity &re is assumed to be constant in an element, rfe and ^ are also constant in the element. The variable transformations
which are useful for suppressing the round-off errors in the calculation, enable the wave eauation G.I55} to be reduced to
3.5
MATRIX ELEMENTS
89
where
3.5
MATRIX ELEMENTS
In this section, we discuss the matrix elements for the first- and secondorder triangular elements. A way to form the global matrixes using matrix elements will be shown in the next section. The eigenvalue equation given in Eqs. (3.155)-(3.158) are
where
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FINITE-ELEMENT METHODS
As mentioned before, since the parameters ql and ^ are constant in each element, only the following terms shown in Eqs. (3.165)-(3.167) have to be calculated to obtain explicit expressions for matrixes [K] and [M]:
Since the shape function [Ne] is expressed by area coordinates L{, L2, and L3, they are useful for performing the integral calculus in Eqs. (3.169)-(3.171). For later convenience, these coordinates are rewritten as
where
and
3.5
MATRIX ELEMENTS
91
Here, Se is the area of the triangle 123 shown in Fig. 3.4. The derivatives of area coordinates L 1? L2, and L3 with respect to coordinates x and y are
The following calculations will use the following convenient integration formula for the area coordinates:
The derivation of this formula is shown in Appendix B. 3.5.1
First-Order Triangular Elements
Figure 3.4 shows a first-order triangular element. Since it has three nodes, its shape function [Ne] has three components:
As shown in Eq. (3.88), we have the following important relations between the shape functions A^, N2, and N3 and the area coordinates L I , L2, and L3:
The next step is to derive the explicit expressions of Eqs. (3.169) to (3.171). A. J $e(d[Ne]/dx)(d[Nef/dx) matrix is
dx dy The component representation of the
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FINITE-ELEMENT METHODS
and the integrand is
Using the integration formula shown in Eq. (3.184), we obtain the following matrix elements:
3.5
B. $$e(d[Ne]/dy)(d[Ne]T/3y) matrix is
dx dy
MATRIX ELEMENTS
93
The component representation of the
and the integrand is
Comparing Eq. (3.187) with Eq. (3.195), we find that the differences between them are the variables for the derivatives. Thus, the elements for
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FINITE-ELEMENT METHODS
the matrix in Eq. (3.195) can be obtained by simply substituting Rf for Qt inEqs. (3.189)-(3.194):
C- JJe[Ne][Ayrd!x dy
and the integrand is
The component representation of the matrix is
3.5
MATRIX ELEMENTS
95
Using the integration formula shown in Eq. (3.184), we obtain the following matrix elements:
We have so far obtained matrixes [Ae], [Be], and [Ce] for the first-order triangular element e. We can form the global matrixes [K] and [M] by substituting matrixes [Ae], [Be], and [Ce] into Eqs. (3.166) and (3.167) and summing them up. Because the matrixes [Ae], [Be], and [Ce] are symmetrical 3 x 3 matrixes, the global matrixes are also symmetrical.
3.5.2
Second-Order Triangular Elements
Since the second-order triangular element (Fig. 3.8) has six nodes, its shape function [Ne] has six components:
As shown in Eqs. (3.135)-(3.140), the important relations between shape functions A ^ , . . . , N6 and area coordinates Ll, L2, and Z,3 are
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FINITE-ELEMENT METHODS
We derive the explicit expressions of Eqs. (3.169)-(3.171) for the second-order triangular element in a way similar to that in which we derived the corresponding expressions for the first-order triangular element. ^- JJe(d[^J/^)(d[A/e]r/d*) dx dy The component representation of the matrix is
and the integrand is
3.5
MATRIX ELEMENTS
97
The derivatives of the shape functions are
Using the integration formula shown in Eq. (3.184) and the relations
we obtain the following matrix elements for Eq. (3.218):
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FINITE-ELEMENT METHODS
3.5
MATRIX ELEMENTS
99
100
FINITE-ELEMENT METHODS
3.5
B. JJe(3[Are]/9y)(3[7Ve]r/ay) dx dy matrix is
MATRIX ELEMENTS
101
The component representation of the
Comparing Eq. (3.219) with Eq. (3.247), we find that the differences between them are the variables for the derivatives. Thus, the elements for the matrix in Eq. (3.247) can be obtained by simply substituting Rt for Qt in Eqs. (3.226H3.246):
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FINITE-ELEMENT METHODS
3.5
^' JJe[^e][^e]r dx dy
MATRIX ELEMENTS
103
The component representation of the matrix is
and the integrand is
Using the integration formula shown in Eq. (3.184), we obtain the following matrix elements:
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FINITE-ELEMENT METHODS
3.6 PROGRAMMING
105
We have thus obtained the matrixes [Ae], [Be], and [Ce] for the secondorder triangular element e. We can form the global matrixes [K] and [M] by substituting [Ae], [Be], and [Ce] into Eqs. (3.166) and (3.167) and summing them. Because [Ae], [Be], and [Ce] are symmetrical 6 x 6 matrixes, the global matrixes are also symmetrical.
3.6
PROGRAMMING
As described above, when using an FEM, we first obtain the matrix elements for element e. We then obtain the global matrixes for the eigenvalue matrix equation by summing the contributions of all the elements. This section discusses how computer programs based on the first- and second-order triangular elements can be written by using
which is the eigenvalue matrix equation. Here, [K], [M], and {(/>} are defined as
106
3.6.1
FINITE-ELEMENT METHODS
First-Order Triangular Elements
Figure 3.10 shows an example of an optical waveguide whose buried structure has been divided into 18 first-order triangular elements 61,..., e18. The core with width W comprises two elements e9 and e10, and Fig. 3.11 shows the local coordinates for element e9. In this figure, the local coordinates of the node numbers 6, 7, and 10—whose coordinates are (x6,y6), (x7, jcy7), and (xio^io)—are respectively 1, 2, and 3. Thus, the node number can be determined from the element number and the local coordinate. The actual programming flow is as follows: 1. Divide the whole analysis region into a number of meshes by using first-order triangular elements.
FIGURE 3.10. Mesh formed by first-order triangular elements.
3.6
PROGRAMMING
107
FIGURE 3.11. Local coordinates for the nodes of a first-order triangular element.
2. As shown in Fig. 3.11, any node number can be identified by specifying the element number and the local coordinate. For example, from the node number of element e^ and the local coordinate 1, we get the node number 7 and the coordinate (x7,y7). 3. Using Eqs. (3.169X3.171), calculate the 3 x 3 matrixes [Ae], [Be], and [Ce] for each element e. 4. Add the calculated results for matrixes [Ae], [Be], and [Ce] to the global matrixes [K] and [M], whose row and column numbers correspond to the combinations of the element numbers and the local coordinates. For example, since the local coordinates 1 and 2 respectively correspond to nodes 7 and 10, the matrix elements with the first row and the second column of [Ae], [Be], and [Ce] are added to the matrix elements of the 7th row and the 10th column of both matrixes [K] and [M]. Since the matrixes used in the scalar FEM are sparse and symmetrical, the amount of memory required for the matrixes can be reduced. 5. As shown in Fig. 3.10, node 6 belongs to six triangular elements (e2, e3, £4, e7, es, and e9). Since each node belongs to more than one triangular element, obtain the global matrixes [K] and [M] by summing the calculated matrix elements of [Ae], [Be], and [Ce] for all triangular elements. 6. When imposing the boundary conditions, which will be discussed in Section 3.7, incorporate them into matrixes [K\ and [M]. 1. Obtain the propagation constant /? or the effective index «eff by solving the eigenvalue matrix equation (3.165).
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FINITE-ELEMENT METHODS
FIGURE 3.12. General meshes formed by first-order triangular elements.
Figure 3.12 shows an example of general meshes formed by first-order triangular elements. The numbers of nodes in the vertical and horizontal directions are respectively My and Mx, which means that the total number of nodes is Mx • My. To discuss sparsity, we consider element el, in which there are three nodes; the numbers are 1, 2, and My + \. Any node whose number is larger than My + 1 is not related to node 1. Figure 3.13 shows matrixes [K] and [M] corresponding to the meshes shown in Fig. 3.12. The physical meanings of the matrix elements can be inferred from the wave functions 4>l, 4>2,..., 4>M and in Fig. 3.13. Matrix element atj (i 7^7) is related to the interaction between wave functions , and (f>p and matrix element au is related to the self-interaction of wave function ,. Thus, as shown in Fig. 3.13, matrixes [K] and [M] have nonzero elements until the My + 1 column. The number My + 1 is called the bandwidth of the sparse matrix.
3.6.2
Second-Order Triangular Elements
Figure 3.14 shows an example of general meshes formed by the secondorder triangular elements, and Fig. 3.15 illustrates the correspondence
3.6
PROGRAMMING
109
FIGURE 3.13. Forms of global matrixes [K] and [M], M = MxMy.
between the node numbers and the local coordinates for element e{. A computer program based on second-order triangular elements is basically the same as one based on the first-order triangular elements, but it should be noted that each second-order triangular element has six local coordinates and that the corresponding matrixes [K] and [M] have a bandwidth of2Mv+l. When the number of the elements is the same, the total number of nodes and the bandwidth are larger for second-order triangular elements than they are for first-order triangular elements. Since the second-order elements approximate the unknown wave functions by quadratic curved surface functions, they are more accurate than the first-order elements, which approximate the unknown wave functions by plane functions. Thus, the second-order elements are numerically more efficient.
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FINITE-ELEMENT METHODS
FIGURE 3.14. General meshes formed by second-order triangular elements.
FIGURE 3.15. Local coordinates for the nodes of a second-order triangular element.
3.7
BOUNDARY CONDITIONS
In this section, we discuss the boundary conditions that should be applied to the nodes on the edges of the analysis region. The eigenvalue matrix equation was shown in Eq. (3.154) as
3.7
BOUNDARY CONDITIONS
111
Here, we discuss the two conditions used most widely.
3.7.1
Neumann Condition
The Neumann condition requires that the derivative of the wave function be set to zero, which means that the variation of the wave function at the boundaries would be negligibly small. Thus, we get
Substituting Eq. (3.300) into (3.299), we get the familiar eigenvalue matrix equation
Here, we mention another important application of boundary conditions. Figure 3.10 shows the whole analysis region. Analyzing the whole region, we simultaneously obtain even modes including a dominant mode and odd modes whose fields are zero at the mirror-symmetrical plane of the structure. On the other hand, we can obtain the solutions for only the even modes or the odd modes by analyzing the half-plane structure (Fig. 3.16) with the Neumann condition or the Dirichlet condition applied at the mirror-symmetrical plane at the center. This is convenient when we analyze the definite modes of optical waveguides.
3.7.2
Dirichlet Condition
The Dirichlet condition requires that the wave functions at the boundaries be set to zero:
Thus, we also get the eigenvalue matrix equation (3.301). The Dirichlet condition requires a further process. Since Eq. (3.302) has to hold for the
112
FINITE-ELEMENT METHODS
FIGURE 3.16. Boundary conditions on a mirror-symmetrical plane.
zth component , of eigenvector {^}, some matrix elements other than the diagonal terms for matrixes [K\ and [M] have to be set to zero:
Since the zth-row elements of the matrixes in Eq. (3.303) satisfy the equation
Eq. (3.302) holds. Another way to implement the Dirichlet condition is to simply omit the nodes at the boundaries because under the Dirichlet condition they do not influence the other nodes. This requires less computer memory than is required when Eq. (3.303) is used.
3.7
BOUNDARY CONDITIONS
113
PROBLEMS 1. In the derivation of the eigenvalue matrix equations (3.40) and (3.56), it was assumed that the wave function $e and its normal derivative rjl d(pe/dn are each continuous at the boundaries of two neighboring elements. This was the basis on which the line integral calculus terms in Eq. (3.51) were canceled out at the boundaries. Discuss the validity of the assumption. ANSWER a. Expq mode:
Thus, at the horizontal interfaces the assumption is valid for both the Expq mode and the E*pq mode. At the vertical interfaces, on the other hand, the wave function
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FINITE-ELEMENT METHODS
Node numbers: 1, 2, 3, ...., 6 Element numbers: e l 5 e2, e^ e* Local coordinates:®, (D, (3) FIGURE P3.1. Simple example of first-order triangular elements.
2. Figure P3.1 shows a simple example of first-order triangular elements, where the total node number is 6. Show the form of the matrix equation for this example. ANSWER
REFERENCES
115
REFERENCES [1] O. C. Zienkiewitz, The Finite Element Method, 3rd ed., McGraw-Hill, New York, 1973. [2] M. Koshiba, H. Saitoh, M. Eguchi, and K. Hirayama, "Simple scalar finite element approach to optical waveguides," IEE Proc. J., vol. 139, pp. 166171, 1992. [3] M. Koshiba, Optical Waveguide Theory by the Finite Element Method, KTK Scientific Publishers and Kluwer Academic Publishers, Dordrecht, Holland, 1992. [4] K. Kawano, S. Sekine, H. Takeuchi, M. Wada, M. Kohtoku, N. Yoshimoto, T. Ito, M. Yanagibashi, S. Kondo, and Y. Noguchi, "4 x 4 InGaAlAs/InAlAs MQW directional coupler waveguide switch modules integrated with spotsize converters and their lOGbit/s operation," Electron. Lett., vol. 31, pp. 96-97, 1995. [5] T. Itoh and R. Mittra, "Spectral domain approach for calculating the dispersion characteristics of microstrip lines," IEEE Trans. Microwave Theory Tech., vol. MTT-21, pp. 496-499, 1973. [6] K. Kawano, T. Kitoh, H. Jumonji, T. Nozawa, M. Yanagibashi, and T. Suzuki, "Spectral domain approach of coplanar waveguide traveling-wave electrodes and their applications to Ti: LiNbO3 Mach-Zehnder optical modulators," IEEE Trans. Microwave Theory Tech., vol. 39, pp. 1595-1601, 1991.
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CHAPTER 4
FINITE-DIFFERENCE METHODS
Since the semivectorial finite-difference methods (SV-FDMs) developed by Stern [1,2] are numerically efficient methods taking polarization into consideration and providing accurate results, they are widely used in the computer-aided design (CAD) software currently available for 2D crosssectional analyses of optical waveguides. Finite-difference schemes are also used in the finite-difference beam propagation methods (FD-BPMs), which are of course also widely used in CAD software and which are discussed in detail in Chapter 5. The present chapter will help readers understand SV-FDMs and their formulation. It will also help readers become familiar with how to program them. Furthermore, it will teach users of CAD software not only how to identify the main causes of errors but also how to decrease the size of errors. When the FEMs were discussed in Chapter 3, the wave equations themselves were not solved, but instead the functional was introduced and the variational principle was used, or the Galerkin method, which is a weighted residual method, was used. The FDMs dealt with in this chapter, in contrast, are more direct approaches to solving the wave equations. They solve eigenvalue matrix equations for electric fields or magnetic fields, equations derived from finite-difference approximations for the wave equations. This chapter briefly describes the finite-difference approximations and then derives the vectorial wave equations. It then obtains the semivectorial wave equations by ignoring the terms for the interaction between two polarized field components in the vectorial wave equations. 117
118
FINITE-DIFFERENCE METHODS
This chapter then discusses the formulation of the SV-FDMs, the errors caused by the finite-difference approximations, and SV-FDM programming. Although Stern's formulation uses equidistant discretization, this chapter uses the more versatile nonequidistant discretization.
4.1
FINITE-DIFFERENCE APPROXIMATIONS
In the FDMs discussed in this chapter, eigenvalue matrix equations are derived by using finite-difference schemes to approximate the wave equations. Let us first briefly examine the finite-difference approximations for the derivatives and then examine the accuracy of these approximations. Assume that a ID function/(%) is continuous and smooth. As shown in Fig. 4.1, the function values/j,^, and^ at* = — /z 1? h2, 0 are expressed as
Next, we can write/i and^ as Taylor series expansions around x — 0:
FIGURE 4.1. Difference approximations for derivatives.
4.1
FINITE-DIFFERENCE APPROXIMATIONS
119
where f^ is an nth derivative defined as
Subtracting Eq. (4.4) from Eq. (4.5), we get an expression for the first derivative:
Thus,
As shown in Eq. (4.7), the error caused by approximating the first derivative at x = 0 with the differential expression
is O(h2} when hl=h2 (equidistant discretization) and is O(h) when h{ ^ h2 (nonequidistant discretization). The error caused by approximating the second derivative with the expression
is also O(h2} when hl = h2 and is O(h) when h} / h2 (see Problem 1). Thus, when CAD software is used, equidistant discretization is preferred whenever there is enough computer memory. When discretization is nonequidistant, we have to be careful that the ratio of h^ to h2 does not change drastically, since a drastic change would greatly increase the size of the errors.
120
4.2 4.2.1
FINITE-DIFFERENCE METHODS
WAVE EQUATIONS Vectorial Wave Equations
In this section, the finite-difference expressions are obtained for the semivectorial wave equations. The approximations used in the process are made clear by starting from the fully vectorial forms of the equations. The vectorial wave equation [Eq. (1.34)] for the electric field E is
Let us consider a structure uniform in the z direction. In this case, the derivative of relative permittivity with respect to z is zero:
Thus, the second term in Eq. (4.10) can be written as
After substituting Eq. (4.12) into Eq. (4.10), we separate Eq. (4.10) into the x and y components. [The derivative with respect to z is indicated in Eqs. (4.13)-(4.30) in the form of 3/3z because that is the formulation used in the beam propagation methods dealt with in Chapter 5.] Thus, we obtain the vectorial wave equation using the electric field components Ex and Ey. Its x component is
and its y component is
4.2
WAVE EQUATIONS
121
Because
and
Eqs. (4.13) and (4.14) can be rewritten as
and
Let us next derive the corresponding components of the equation for the magnetic field H. The vectorial wave equation [Eq. (1.40)] for the magnetic field H is
Here, the second term in Eq. (4.19) is investigated in detail. Recall that we are considering a structure uniform in the z direction and that Eq. (4.11) therefore holds. When i, j, and k are respectively assumed to be unit vectors in the x, y, and z directions, we can obtain
122
FINITE-DIFFERENCE METHODS
by assuming that dsr/dz = 0 and using the expressions
The substitution of Eqs. (4.21), (4.22), and (4.23) into Eq. (4.20) results in
Substituting Eq. (4.24) into Eq. (4.19) and separating the result into tin x and y components, we obtain the vectorial wave equation using th magnetic field components Hx and Hv. Its jc component is
and its y component is
And because
and
4.2 WAVE EQUATIONS
123
Eqs. (4.25) and (4.26) can be rewritten as
and
Furthermore, since what we are concerned with here is a 2D crosssectional analysis, the derivatives of the electric and magnetic fields with respect to z are constant:
where ft is a propagation constant. Thus, using Eqs. (4.13) and (4.14), we obtain for the x component
and for the y component
They can be rewritten as
and
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FINITE-DIFFERENCE METHODS
The last terms in Eqs. (4.32)-(4.35) correspond to the interactions between the jc-directed electric field component Ex and the ^-directed electric field component Ey. Using Eqs. (4.25) and (4.26), on the other hand, we obtain for the x component
and for the y component
which can be rewritten as
and
Similar to what we saw in the corresponding equations for the electric field E, the last terms in Eqs. (4.36)-(4.39) correspond to the interactions between the x-directed magnetic field component Hx and the j-directed magnetic field component Hy.
4.2.2
Semivectorial Wave Equations
In equations for the fields that propagate in optical waveguides, the terms corresponding to the interaction between the x-directed electric field component Ex and the j-directed electric field component Ey,
4.2
WAVE EQUATIONS
125
and the terms corresponding to the interaction between the jc-directed magnetic field component Hx and the y-directed magnetic field component Hy,
are usually small. Ignoring these terms for the interaction, we can decouple the vectorial wave equations for the x- and ^-directed field components and reduce them to semivectorial wave equations, which can be solved in a way that is numerically efficient. Semivectorial analyses that neglect the terms for the interaction are therefore widely used when designing optical waveguide devices for which the coupling between the x- and jy-directed polarizations does not have to be taken into consideration. As shown in Fig. 4.2, these analyses can be divided into the quasi-TE mode analysis, in which the principal field component is Ex or Hy, and the quasi-TM mode analysis, in which the principal field component is Ey orHx.
FIGURE 4.2. Principal field components for (a) quasi-TE and (b) quasi-TM modes.
126
FINITE-DIFFERENCE METHODS
A. Quasi-TE Mode The principal field component in the electric field representation for the quasi-TE mode, where the j-directed electric field component Ey is assumed to be zero, is the x-directed electric field component Ex. So according to Eq. (4.32), the semivectorial wave equation for the quasi-TE mode is
which, according to Eq. (4.34), can be rewritten as
The principal field component in the magnetic field representation for the quasi-TE mode, on the other hand, where the x-directed magnetic field component Hx is assumed to be zero, is the j-directed magnetic field component Hy. So according to Eq. (4.37), the semivectorial wave equation is
which, according to Eq. (4.39), can be rewritten as
B. Quasi-TM Mode The principal field component in the electric field representation for the quasi-TM mode, where the x-directed electric field component Ex is assumed to be zero, is the j-directed electric field component Ey. According to Eq. (4.33), the semivectorial wave equation for the quasi-TM mode is therefore
4.3
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
127
which, according to Eq. (4.35), can be rewritten as
The principal field component for the quasi-TM mode in the magnetic field representation, where the y-directed magnetic field component Hy is assumed to be zero, is the x-directed magnetic field component Hx. As before, the semivectorial wave equation based on Eq. (4.36) is
which, according to Eq. (4.38), can be rewritten as
4.2.3
Scalar Wave Equation
In the vectorial and semivectorial wave equations discussed above, the derivatives of relative permittivity &r with respect to the x and y coordinates are taken into consideration. If we assume these derivatives to be zero, or that
the wave equations can be reduced to the scalar wave equation
where (/> is a wave function and designates a scalar field.
4.3 FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS We can derive the finite-difference expressions for the semivectorial wave equations by using the finite differences discussed in Section 4.1 to approximate the derivatives of the semivectorial wave equations for the
128
FINITE-DIFFERENCE METHODS
electric field or the magnetic field representations for the quasi-TE and quasi-TM modes. The currently available CAD software for 2D crosssectional analyses and 3D beam propagation analyses makes use of the finite-difference expressions discussed in this section. Figure 4.3 shows a discretization used in a 2D cross-sectional analysis of optical waveguides. Here, the pair ( p , q} is assumed to correspond to the (x, y) coordinates of a node. It should be noted that the interface of two materials is set midway between two nodes in order to minimize the error caused by the difference approximation. Although the discretization used by Stern [1, 2] was an equidistant one, the more versatile scheme described here is nonequidistant, with discretization widths e and w in the x direction and n and s in the y direction. The scalar case is also briefly discussed here. Although the variable transformations x = xk0 and y = ykQ are useful for reducing the round-off error in an actual calculation, they are not used here for simplicity. Reasonable results for the facet reflectivities of 3D semiconductor optical waveguides [3] have been obtained by incorporating the semivectorial finite-difference expressions discussed in this section into the bidirectional method of line BPM (MoL-BPM) [4].
4.3.1
Quasi-TE Mode
A. Ex Representation The Ex representation wave equation derived for the quasi-TE mode [Eq. (4.40)] is
FIGURE 4.3. Nonequidistant discretization for the finite-difference method.
4.3
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
129
What we want to do now is derive the finite-difference expression for this equation. As shown in Fig. 4.3, the field components and coordinates at nodes are expressed as
Using for Ep+lq, Ep_^q, Ep^q+l, and Ep^q_v Taylor series expansions around (p,q), we get
First, we derive the finite-difference expression for the first term in Eq. (4.50). Multiplying Eq. (4.60) by w and Eq. (4.61) by e and then adding them, we find that
130
FINITE-DIFFERENCE METHODS
Therefore
and
As shown in Section 4.1, the error caused by the finite-difference approximation in Eq. (4.65) is O(e — w) when e ^ w (nonequidistant discretization) and is 0((Ax)2) when e = w = Ax (equidistant discretization). We use a similar procedure to obtain the finite-difference expression for the derivative with respect to y [the third term in Eq. (4.50)]. Multiplying Eq. (4.62) by n and Eq. (4.63) by s and adding them, we get
Therefore
4.3
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
131
and
According to Eq. (4.66), the error caused by the finite-difference approximation in Eq. (4.67) is O(s — ri) when n ^ s (nonequidistant discretization) and is 0((Ay)2) when s = n = Ay (equidistant discretization). Now, let us consider the second term in Eq. (4.50), which includes a derivative of relative permittivity er For simplicity here, expression (4.6) is used to represent derivatives such as d/dx and 92/3;c2. The difference center of the equations under discussion is (p,q). Here, we introduce ( p + ^, q) as a hypothetical difference center between nodes (/?, q) and (p+ l,q) and introduce (p — ^, q) as a hypothetical difference center between nodes (p — \,q) and (p,q). Using these two hypothetical difference centers, we can get the following two Taylor series expansions:
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FINITE-DIFFERENCE METHODS
Subtracting Eq. (4.69) from Eq. (4.68), we get
Thus, the first derivative is
Finally, we get
According to Eq. (4.70), the error caused by the finite-difference approximation in Eq. (4.71) is O(e — w) when e ^ w (nonequidistant discretization) and is <9((Ax)2) when e = w — Ax (equidistant discretization). The next step is to derive the finite-difference expression for Eq. (4.71), and this is done by calculating the two terms within the brackets on the right-hand side.
4.3
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
133
Calculating Taylor series expansions of Ep+l>q and Ep^q around the hypothetical difference center ( p +1, q\ we get
Adding Eq. (4.72) to Eq. (4.73), we get
Therefore
Using the expression sr(p, q) for sr(xp,yq) and calculating Taylor series expansions of &r(p + 1, q) and £r(p, q) around the hypothetical difference center ( p + \, q\ we get
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FINITE-DIFFERENCE METHODS
And adding Eq. (4.75) to Eq. (4.76), we get
Therefore
On the other hand, subtracting Eq. (4.76) from Eq. (4.75), we get
Thus
Combining Eqs. (4.77) and (4.78), we get
4.3
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
135
Finally, we can derive the following equation by using Eqs. (4.74) and (4.79):
Now that we have the first term within the brackets on the right-hand side of Eq. (4.71), we can derive the second term by using the same procedure. Calculating Taylor series expansions ofEpq and Ep_{^q around the hypothetical difference center ( p — \, q\ we get
Adding Eq. (4.81) to Eq. (4.82), we get
Therefore
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FINITE-DIFFERENCE METHODS
Similarly, calculating Taylor series expansions of &r(p — \,q) and er(p, q) around the hypothetical difference center (p — |, q), we get
Adding Eq. (4.84) to Eq. (4.85) gives
whereas subtracting Eq. (4.85) from Eq. (4.86) gives
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FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
137
Combining Eqs. (4.86) and (4.87), we get
Finally, we can derive the following equation by using Eqs. (4.83) and (4.88):
Now that we have obtained the two terms within the brackets on the right-hand side of Eq. (4.71) by using Taylor series expansions, let us further our understanding by deriving equations for these terms without using Taylor series expansions. With respect to the first term, the difference center is (p + ^,q). To set the difference center at (p +1, q), we need to take the average of (p + 1, q) and (p, q). Thus, we get the relations
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FINITE-DIFFERENCE METHODS
We can immediately derive the equation
With respect to the second term, on the other hand, the hypothetical difference center is (p — |, q). To set the difference center at ( p — |, q), we need to take the average of (/>, q) and (p — 1, q). Thus, we get the relations
Again we can immediately derive the equation
We can see that Eqs. (4.93) and (4.97) are equivalent to Eqs. (4.80) and (4.89). Let us return here to the main topic. Substituting Eqs. (4.80) and (4.89) into Eq. (4.71), we get
4.3
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
Substituting Eqs. (4.65), (4.67), and (4.98) into Eq. (4.50), we get
139
140
FINITE-DIFFERENCE METHODS
Thus, we get the following finite difference expression for Eq. (4.50):
where
f. Hy Representation The Hy representation wave equation derived >r the quasi-TE mode [Eq. (4.43)] is
Now, let us derive the finite-difference expression for this equation. To do this, we start with the first term of Eq. (4.106). Again using (p + ^ , q) as the hypothetical finite-difference center between nodes (/>, q) and
4.3
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
141
( p + 1, q) and using ( p — ^, q) as the hypothetical finite-difference center between nodes (p — l,q) and (p,q\ we get
Subtracting Eq. (4.107) from Eq. (4.106), we get
Thus,
142
FINITE-DIFFERENCE METHODS
Finally, we obtain
According to Eq. (4.109), the error caused by the finite-difference approximation in Eq. (4.110) is O(e — w) when e ^ w (nonequidistant discretization) and is 0((Ax)2) when e = w = Ax (equidistant discretization). The next step is to derive the finite-difference expression for Eq. (4.110) by calculating the two terms within the brackets on the righthand side. Now, we calculate the first term. Calculating Taylor series expansions of Hp+l^q and Hp
Subtracting Eq. (4.112) from Eq. (4.111), we get
Therefore
4.3
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
143
Recall that the relative permittivity at (p + \, q) [Eq. (4.77)] is
Thus, according to Eqs. (4.113) and (4.114),
Next, we calculate the second term within the brackets in Eq. (4.110). Calculating Taylor series expansions of Hpq and Hp_l^q around the hypothetical difference center ( p — ^, q), we get
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FINITE-DIFFERENCE METHODS
Subtracting Eq. (4.117) from Eq. (4.116), we get
Therefore
Recalling that the relative permittivity at (p — ^, q) [Eq. (4.86)] is
and using Eqs. (4.118) and (4.119), we get
We now have the finite-difference expressions for the first and second terms of Eq. (4.110). Although it is not shown here, these finite-difference expressions can be derived more easily with the procedures used in deriving Eqs. (4.93) and (4.97) (see Problem 3).
4.3
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
145
Substitution of Eqs. (4.115) and (4.120) into Eq. (4.110) results in
Thus, we get the finite-difference expression
On the other hand, the finite-difference expression for the derivative with respect to y [Eq. (4.67)] is
Here, the error caused by the finite-difference approximation in Eq. (4.122) is O(s — ri) when n ^ s (nonequidistant discretization) and is 0((Ay)2) when s = n = Ay (equidistant discretization).
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FINITE-DIFFERENCE METHODS
Substituting Eqs. (4.121) and (4.122) into Eq. (4.106), we get the following finite-difference expressions for Eq. (4.106):
Thus, we get the following final finite-difference expression for Eq. (4.106):
4.3
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
147
where
It should be noted that since the forms of the derivatives with respect to x differ slightly between the wave equations of the electric field and magnetic field representations, the resultant finite-difference expressions for aw, y.e, and &x also differ between the two representations. 4.3.2
Quasi-TM Mode
A. Ey Representation The Ey representation wave equation for the quasi-TM mode [Eq. (4.44)] is
Comparing this equation with the Ex representation wave equation for the quasi-TE mode [Eq. (4.50)], we find that we can obtain the finitedifference expression for the Ey representation for the quasi-TM mode [Eq. (4.130)] by making the following replacements in Eqs. (4.100)(4.105):
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FINITE-DIFFERENCE METHODS
Thus, we get the following finite-difference expression for Eq. (4.130):
where
B. Hx Representation The Hx representation wave equation for the quasi-TM mode [Eq. (4.43)] is
Comparing this equation with the Hy representation for the quasi-TE mode [Eq. (4.106)], we find that the finite-difference expression for the Hx representation for the quasi-TM mode [Eq. (4.140)] can be obtained by making the following replacements in Eqs. (4.124)-(4.129):
4.3
FINITE-DIFFERENCE EXPRESSIONS OF WAVE EQUATIONS
149
Thus, we get the following finite-difference expression for the Hx representation:
where
4.3.3
Scalar Mode
The scalar wave equation [Eq. (4.49)] is
Since the derivatives of relative permittivity with respect to :c or y in the semivectorial wave equations are set to zero for the scalar analysis, we can
150
FINITE-DIFFERENCE METHODS
easily get the finite-difference expression for the scalar finite-difference method (SC-FDM):
where
4.4
PROGRAMMING
Now, we look at how an eigenvalue matrix equation can be solved using the FDM. The procedure is almost the same as that for the solution using the FEM, described in detail in Chapter 3, so only important differences are dealt with here. The finite-difference expression for the semivectorial wave equation was obtained by approximating the derivatives with the difference expressions:
where (p, q) corresponds to coordinates (x,y) and where (j)p^q corresponds, for the quasi-TE mode, to the field component Ex or Hy and corresponds, for the quasi-TM mode, to the field component Ey or Hx.
4.4
PROGRAMMING
151
FIGURE 4.4. Meshes for the finite-difference method.
Figure 4.4 shows a mesh model in a finite-difference scheme in which the whole analysis area is divided into a number of meshes and the nodes are numbered from top to bottom and from left to right. Calculating Eq. (4.158) for each node in an Mx x My matrix, we obtain the following eigenvalue matrix equation:
Here, f>2 is an eigenvalue and {0} is an eigenvector expressed as
where M is Mx x My. It should be noted that, if the variable transformations x = xkQ and y = yk0 mentioned in Section 4.4 are used, the eigenvalue in Eq. (4.159) is n^ff, where «eff(/?Ao) is me effective index. Figure 4.5 shows the global matrix [A] corresponding to the meshes shown in Fig. 4.4.
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FINITE-DIFFERENCE METHODS
FIGURE 4.5. Form of global matrix [A]. Here, M = MxMy.
We can obtain the propagation constant and field distribution by solving the eigenvalue matrix equation (4.159). In the SV-FDMs, the global matrix [A] is a nonsymmetric sparse matrix. In the SC-FDM, on the other hand, the global matrix is symmetric, so only half of it has to be calculated. Taking these facts into account and considering the interaction between nodes shown in Figs. 4.4 and 4.5, we can easily understand that the bandwidth of the global matrix is 2My + 1 in the SV-FDM and is My + 1 in the SC-FDM. The latter bandwidth is the same as that of the global matrix in the first-order SC-FEM. In the actual programming, the node number r is used instead of (/?,#). When nodes are numbered from top to bottom and from left to right as shown in Fig. 4.4, the node number r for (p, q) can be designated as follows:
where 1 < p < Mx and 1 < q < My. The rth row in the eigenvalue matrix equation can be expressed as
4.5
BOUNDARY CONDITIONS
153
where atj is an element of the global matrix [A]. The coefficients of Eq. (4.158) and Eq. (4.162) correspond as follows:
The eigenvalue matrix equation is formed in basically the same way as that used to form the eigenvalue matrix equation for the FEM discussed in Section 3.6. There is, however, one important difference. In the FEM, the whole analysis area is divided into a number of elements, and the variational principle or the Galerkin method is applied to each element. Therefore, as shown in Eqs. (3.293) and (3.294), when the FEM is used, the formation of the global matrix requires a summation over all the elements. In the FDM, on the other hand, the derivatives in the wave equations are approximated with finite differences. The FDM thus does not require a summation in order to form the global matrix.
4.5
BOUNDARY CONDITIONS
In the actual programming, we have to impose boundary conditions on the nodes on the edge of the analysis window. In other words, it is necessary that the effect of nodes outside the analysis window be taken into account at the edge of the window. Here, the Dirichlet, Neumann, and analytical boundary conditions will be discussed. DIRICHLET CONDITION set to zero. Thus,
A wave function outside the analysis window is
NEUMANN CONDITION The normal derivative of a wave function at the edge of the analysis window is set to zero. Thus,
154
FINITE-DIFFERENCE METHODS
In other words, it is assumed that the value of a hypothetical wave function outside the analysis window is equal to that of an actual wave function at the edge of the analysis window. ANALYTICAL BOUNDARY CONDITION When we assume that the wave number in a vacuum, the effective index, the discretization width at the edge of the analysis window, and the relative permittivity are respectively #o, «eff, A and er(p, q\ the analytical wave function outside the analysis window to be connected with a wave function at the edge of the analysis window is assumed to decay exponentially with the decay constant
-koV\nlff-£r(P,
WRITING BOUNDARY CONDITIONS INTO PROGRAMS Let us consider here how the boundary conditions can be written into a computer program. Although Eq. (4.162) has to be used for programming, for simplicity we will instead consider Eq. (4.158), written here as
a. Left-Hand Boundary (p = 1 and except at corners) Consider the left-hand boundary shown as 1 in Fig. 4.6. Here, the discretization width is Ax. When we assume that (p, q} is a node on the boundary, the hypothetical node outside the analysis window is ( p — 1, q} and we get
For the Dirichlet, Neumann, and analytical conditions, the coefficient yL is expressed as
4.5
BOUNDARY CONDITIONS
155
FIGURE 4.6. Nodes on boundaries and hypothetical nodes outside boundaries.
Substituting Eqs. (4.172) and (4.173) into Eq. (4.171), we get
b. Right-Hand Boundary (p = Mx and except at corners) Consider the right-hand boundary shown as 2 in Fig. 4.6. Here, the discretization width is again Ax. When we assume that ( p , q ) is a node on the boundary, the hypothetical node outside the analysis window is (p+ 1, #) and we get
For the Dirichlet, Neumann, and analytical conditions, the coefficient yR is expressed as
156
FINITE-DIFFERENCE METHODS
Substituting Eqs. (4.175) and Eq. (4.176) into Eq. (4.171), we get
c. Top Boundary (q = 1 and except at corners) Consider the top boundary shown as 3 in Fig. 4.6. Here, the discretization width is Ay. When we assume that (p,q) is a node on the boundary, the hypothetical node outside the analysis window is ( p , q — 1) and we get
For the Dirichlet, Neumann, and analytical conditions, the coefficient yv is expressed as
Substituting Eqs. (4.178) and (4.179) into Eq. (4.171), we get
d. Bottom Boundary (q = My and except at corners) Consider the bottom boundary shown as 4 in Fig. 4.6. Here, the discretization width is again Ay. When we assume that (p, q) is a node on the boundary, the hypothetical node outside the analysis window is (p, q + 1) and we get
For the Dirichlet, Neumann, and analytical conditions, the coefficient yD is expressed as
4.5
BOUNDARY CONDITIONS
157
Substituting Eqs. (4.181) and (4.182) into Eq. (4.171), we get
e. Left-Top Corner (p = q = \) Consider the left-top corner shown as 5 in Fig. 4.6. Here, the discretization widths are Ax and Ay. When we assume that (/?, q} is the node at the corner, the hypothetical nodes outside the analysis window are (p — l,q) to the left and ( p , q — 1) to the top. We thus get
For the Dirichlet, Neumann, and analytical, conditions, the coefficients yL and jv are expressed as
Substituting Eqs. (4.184) to (4.187) into Eq. (4.171), we get
/ Left-Bottom Corner (p=\,q = My) Consider the left- bottom corner shown as 6 in Fig. 4.6. Here, the discretization widths are again Ax and Ay. When we assume that ( p , q} is the node at the corner, the hypothetical nodes outside the analysis window are ( p — 1, q) to the left and (/?, q + 1) to the bottom. We thus get
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FINITE-DIFFERENCE METHODS
For the Dirichlet, Neumann, and analytical conditions, the coefficients yL and JD are expressed as
Substituting Eqs. (4.189) to (4.192) into Eq. (4.171), we get
g. Right-Top Corner (p — Mx, q = 1) Consider the right-top corner shown as 7 in Fig. 4.6. Here, the discretization widths are again Ax and Ay. When we assume that (p, q) is the node at the corner, the hypothetical nodes outside the analysis window are ( p + 1, q) to the right and (p, q — 1) to the top. We thus get
For the Dirichlet, Neumann, and analytical conditions, the coefficients yR and yv are expressed as
4.5
BOUNDARY CONDITIONS
159
Substituting Eqs. (4.194)-(4.197) into Eq. (4.171), we get
h. Right-Bottom Corner (p = Mx, q = My) Consider the right-bottom corner shown as 8 in Fig. 4.6. Here, the discretization widths are again Ax and Ay. When we assume that (p, q} is the node at the corner, the hypothetical nodes outside the analysis window are ( p + 1, q) to the right and (/?, q + 1) to the bottom. We thus get
For the Dirichlet, Neumann, and analytical conditions, the coefficients yR and yD are expressed as
Substituting Eqs. (4.199)-(4-202) into Eq. (4.171), we get
In a way similar to that described in Section 3.7 for the FEM, we can calculate the even or the odd mode by assuming Neumann or Dirichlet conditions at the symmetry plane. This increases the numerical efficiency in terms of central processing unit (CPU) time and computer memory.
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FINITE-DIFFERENCE METHODS
4.6
NUMERICAL EXAMPLE
The finite-difference expressions in this chapter are used in CAD software currently available on the market. Readers can also develop software by themselves. This section briefly discusses a calculation model and results calculated using SV-FDM software. Figure 4.7 shows a calculation model that has a 0.4-|o.m2 core. The refractive indexes for a wavelength of 1.55 jim are 3.5 for the core and 3.1693 for the cladding. The nonequidistant discretization scheme was used in this example. The number of nodes Mx in the horizontal direction and the number of nodes My in the vertical direction were both 96, and the minimum and maximum discretization widths in both directions were respectively 0.025 and 0.05 um. The calculated effective index neff for both the quasi-TE and quasi-TM modes was 3.2172, and Fig. 4.8 is a three-dimensional plot of the electric field component Ex calculated for the quasi-TE mode. It should be noted that since the normal component of the electric flux density Dx = srEx is continuous at the interface between two media, as shown in Eq. (1.55), the normal component of the electric field Ex is not continuous at the interface. In Fig. 4.8, we can clearly see the discontinuities of Ex. Since, as pointed out in Sections 4.1 and 4.3, the finite-difference expressions have the errors of the order of h for nonequidistant discretization and of the order of h2 for equidistant discretization, equidistant discretization is preferable. But because extremely fine meshes are necessary for dealing with interfaces at which the refractive index changes abruptly, as in this calculation model, nonequidistant discretization must
FIGURE 4.7. Calculation model.
PROBLEMS
161
FIGURE 4.8. The ^-directed electric field component Ex.
be used in order to reduce the amounts of computer memory and CPU time required. The scalar FEM did not give the degenerated results for the square core due to inaccurate boundary conditions, as discussed in Chapter 3. The semivectorial FDMs, on the other hand, can overcome this difficulty for a small-aspect core structure. PROBLEMS 1. Show that when the second derivative is approximated by Eq. (4.9),
the error is O(h2) when h{ = h2 (equidistant discretization) and is O(H) when hl ^ h2 (nonequidistant discretization). ANSWER Multiplying Eq. (4.4) by h2 and Eq. (4.5) by hl and adding them, we get
from which the answer can easily be obtained.
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FINITE-DIFFERENCE METHODS
2. Derive the finite-difference expression (4.121) without using the Taylor series expansion.
ANSWER Since the hypothetical difference center is ( p + \, q) for the first term within the brackets on the right-hand side of Eq. (4.110), we can derive the equations
and
from which we can immediately derive
And since the hypothetical difference center is (p — \,q) for the second term within the brackets, we can derive the equations
and
from which we can immediately derive
Substituting Eqs. (P4.5) and (P4.8) into Eq. (4.110) and using some mathematical manipulations, we can easily derive Eq. (4.121).
PROBLEMS
163
FIGURE P4.1. Simple example of the finite-difference method.
3. A simple calculation model for the semivectorial FDM is shown in Fig. P4.1. Show the form of the global matrix [A] of Eq. (4.159).
ANSWER The global matrix [^4] is shown in Fig. P4.2. It should be noted that the interactions between nodes 3 and 4, which correspond to a3 4 and a4>3, and the interactions between nodes 6 and 7, which correspond to a6 7 and a7>6, have to be set to zero. Since the number of nodes in the^ direction is 3, the bandwidth is 7. The reason for this is explained in Section 4.4. 4. Calculate effective indexes neff for the quasi-TE and quasi-TM modes of the strip-loaded optical waveguide shown in Fig. P4.3. Assume that the refractive indexes for a wavelength of 1.55 um are 3.3884 for the InGaAsP of the 1.3-um band-gap wavelength (1.30 core and 3.1693 for the InP cladding.
FIGURE P4.2. Form of global matrix [A].
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FIGURE P4.3. Calculation example of a strip-loaded optical waveguide.
ANSWER The results you get will slightly depend on the CAD software used to calculate them, but the effective index for the quasi-TE mode is 3.2576 and that for the quasi-TM mode is 3.2471.
REFERENCES [1] M. Stern, "Semivectorial polarized finite difference method optical waveguides with arbitrary index profiles," lEEProc. J., vol. 135, pp. 56-63, 1988. [2] M. Stern, "Semivectorial polarized H field solutions for dielectric waveguides with arbitrary index profiles," IEE Proc. J., vol. 135, pp. 333-338, 1988. [3] K. Kawano, T. Kitoh, M. Kohtoku, T. Takeshita, and Y. Hasumi, "3-D semivectorial analysis to calculate facet reflectivities of semiconductor optical waveguides based on the bi-directional method of line BPM (MoL- BPM)," IEEE Photon. Technol. Lett., vol. 10, pp. 108-110, 1998. [4] B. Gerdes, B. Lunitz, D. Benish, and R. Pregla, "Analysis of slab waveguide discontinuities including radiation and absorption effects," Electron. Lett., vol. 28, pp. 1013-1015, 1992.
CHAPTER 5
BEAM PROPAGATION METHODS
The analysis methods discussed in the preceding chapters assumed the structures of the optical waveguides to be uniform in the propagation direction. A lot of the waveguides used in actual optical waveguide devices, however, have nommiform structures such as bends, tapers, and crosses in the propagation direction. In this chapter, we discuss the beam propagation methods (BPMs) that have been developed for the analysis of such nommiform structures. Various kinds of BPMs, such as the fast Fourier transform (FFT-BPM) [1-4], the finite difference (FD-BPM) [4-11], and the finite element (FEBPM) [12], have been developed. For the derivatives with respect to the coordinates in the lateral directions, they respectively make use of the fast Fourier transform (FFT), the finite-difference (FD) approximation, and the finite-element (FE) approximation. The FFT-BPM and the FD-BPM will be discussed here. Beam propagation CAD software is widely available on the market.
5.1 FAST FOURIER TRANSFORM BEAM PROPAGATION METHOD The FFT-BPM [1] had been widely applied to design optical waveguides until the FD-BPM [4] was developed. The FFT-BPM has the following disadvantages due to the nature of the FFT: (1) it requires a long 165
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computation time, (2) the discretization widths in the lateral directions must be uniform, (3) the simple transparent boundary condition cannot be used at the analysis boundaries, (4) very small discretization widths cannot be used in the lateral directions, (5) the polarization cannot be treated, (6) it is inadequate for large-index-difference optical waveguides, (7) the number of sampling points must be a power of 2, and (8) the propagation step has to be small. But it is investigated here because the FFT-BPM is historically important and the line of thinking it exemplifies is very interesting and useful. 5.1.1
Wave Equation
The scalar Helmholtz equation is expressed as
where V is the Laplacian
and &0 is the wave number in a vacuum. Here, the slowly varying envelope approximation (SVEA) is used to approximate the wave function i[/(x, y, z) of the light propagating in the +z direction. In this approximation, \j/(x,y,z) is separated into the slowly varying envelope function (f)(x, y, z) and the very fast oscillatory phase term exp(—j$z) as follows:
Here,
where neff is the reference index, for which the refractive index of the substrate or cladding is usually used. Substituting the second derivative of the wave function \f/(x, y, z) with respect to z,
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into Eq. (5.1) and dividing both sides by the exponential term exp(—y'/fe), we get
where V^_ is a Laplacian in the lateral directions (i.e., the x and y directions) and is expressed as
Or, using the relation
we get
Since the second derivative of the wave function (/> with respect to z is not neglected, Eq. (5.9) is a wide-angle formulation. On the other hand, when the second derivative is neglected, that is, when we assume
the wave equation (5.9) is reduced to
The assumption that the second derivative of the wave function (j) with respect to z can be neglected is called the Fresnel approximation or the para-axial approximation. The equality of the Fresnel approximation to the para-axial approximation will be discussed in Section 5.1.5.
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5.1.2
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Fresnel Approximation
Let us try to solve the wave equation (5.11), which is based on the Fresnel approximation. First, we separate the variables of the wave function (f> of the Fresnel wave equation into the propagation direction and the lateral directions:
Substituting Eq. (5.12) into Eq. (5.11) and dividing both sides by , we get
and therefore
Substituting y in Eq. (5.13) into Eq (5.12), we get
Thus, the wave function (j)(x, y,z + Az), which advances further than (f)(x, y, z) by Az in the propagation direction, can be written as
We separate the exponential term into the following two terms:
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where
Here, we used the relation n — «eff + An and, assuming that An is sufficiently small, neglected (An)2. Since V^ in Eq. (5.16) is a derivative operator, the following relation holds for the general function/:
This relation implies that the first and second operators of Eq. (5.16) cannot be interchanged (i.e., they are not commutable). However, we symmetrize the operators in Eq. (5.16) as
Although the reasons are not discussed here because of space limitations, Eq. (5.19) results in errors of the order of (Az)3. This is one of the reasons that the propagation step Az in the FFT-BPM has to be small. It should also be pointed out that Eq. (5.16), which generates the unsymmetrical operators, results in larger errors: errors of the order of (Az)2 [2]. Now, let us discuss the physical meaning of each term in Eq. (5.19). When the refractive index is assumed to be uniform in the analysis region, X is equal to zero. Thus, the wave equation Eq. (5.19) is reduced to
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This implies that the operator
corresponds to the propagation over Az in free space. Therefore, the first and the third terms of Eq. (5.19) correspond to the free-space propagation of the light over Az/2. Thus, Eq. (5.19) implies that the wave function at z + Az can be obtained by first advancing the wave function at z by Az/2 in free space, then giving a phase shift (—/) due to a phase-shift lens, and finally advancing the wave function by another Az/2 in free space. Next, we will obtain the explicit expression of the free-space propagation operator (5.21) by actually applying it to the wave function. The discrete Fourier transform (i.e., the spectral domain wave function) is
where
Here, X and Y are the widths in the x and y directions. The inverse discrete Fourier transform, on the other hand, is
where the coefficient l/MN is omitted for simplicity. According to the operator (5.21) and the above discussion, when the wave propagates over Az/2 in free space, we get the following wave function at z + Az/2:
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We can get the wave function at z + Az/2 by replacing z by z + Az/2 in Eq. (5.24):
On the other hand, substituting the right-hand side of Eq. (5.24) for 4>(x,y,z) on the left-hand side of Eq. (5.25), we get
As the right-hand side of this equation can be rewritten as
we get another expression for the wave function at z + Az/2:
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Since the wave functions (#,y, z + Az/2) given in Eqs. (5.26) and (5.28) have to be equal to each other, we get the important relation
Equation (5.29) shows the relation between the spectral domain wave function $mn(z + Az/2) at z + Az/2 and the spectral domain wave function (f)mn(z) at z. The exponential term on the right-hand side of Eq. (5.29),
corresponds to the propagation over Az/2 in free space. We also find that Eq. (5.28) is the inverse discrete Fourier transform of the function
From these discussions, we draw the conclusion that the application of the operator
which corresponds to the propagation over Az/2 in free space, to the space-domain wave function
to the space-domain wave function (/>(*, 7, z). Here, the symbols X and X"1 respectively represent the discrete Fourier transform and the inverse discrete Fourier transform. The variables kx and ky are expressed as
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FIGURE 5.1. Calculation of one period in the FFT-BPM.
Thus, the FFT-BPM calculation procedures for a period Az can be summarized as follows, where steps 1-5 correspond to the labels in Fig. 5.1: 1. At the propagation position z, calculate the spectral domain wave function (j>mn(z) in the Fourier transform domain by taking the Fourier transform of the space-domain wave function (f)(x, y, z). 2. To get the transformed wave function $mn(z + Az/2) at z + Az/2, multiply
by the spectral domain wave function 4>mn(z) obtained in step 1. This multiplication corresponds to the propagation over the distance Az/2 in free space. 3. Taking the inverse Fourier transform of the spectral domain wave function (j)mn(z + Az/2) obtained in step 2, obtain the space-domain wave function (f)(x, y,z + Az/2) just in front of the phase-shift lens. Then, multiplying the phase-shift term exp(-j%) due to the phaseshift lens by the space-domain wave function (j>(x, y,z + Az/2), obtain the space-domain wave function just after the phase-shift lens:
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4. Taking the Fourier transform of the space-domain wave function just after the phase-shift lens and multiplying it by
corresponding to the propagation over Az/2 in free space, obtain the spectral domain wave function $mn(z + Az) at z + Az. 5. When the space-domain wave function (x, y, z + Az) at z + Az is necessary, take the inverse Fourier transform of the spectral domain wave function (j)mn(z + Az) obtained in step 4. Repeating steps 1-5, we can get the space-domain wave function at the target propagation position. It should be noted that if the space-domain wave function at each z + Az is not necessary, one should return directly to step 2 from step 4 and repeat steps 2-4. 5.1.3
Wide-Angle Formulation
Up to this stage (i.e., in the Fresnel approximation), the second derivative of the slowly varying envelope function (/> with respect to z has been neglected. Now, we return to the wide-angle wave equation (5.9), which contains the second derivative. Readers who feel they have no need for such a discussion of the wide-angle formulation for the FFT-BPM can skip this section. Substituting the wave function given by Eq. (5.12) into Eq. (5.9) and dividing both sides by , we get
Therefore
and
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Since the wave is supposed to propagate in the +z direction, the minus sign in Eq. (5.36) should be used so that
Substituting this y into Eq. (5.12), we get the wave function at z:
Finally, we get the wave function at z + Az:
Now, we expand the second term inside the exponential function of Eq. (5.39):
The Laplacian V^ in the lateral directions corresponds to the square of the wave numbers in the lateral directions. The wave numbers in the lateral directions are much smaller than the wave number in the propagation direction (i.e., the propagation constant). Thus, the assumption V^ <$C /?2 is concluded to be valid. In addition, taking (n2 — n2^} = (n + neff)(n — « eff ) ^ 2«eff(« — w eff ) and /? = £0weff into consideration, we can simplify the right-hand side of Eq. (5.40) to
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Substituting Eq. (5.41) into Eq. (5.39), we get
Since the propagation constant ft has the order of the inverse of the wavelength, it is a large number. Inside the exponential function, (ft2 + V^)1/2, which is also the large number, is subtracted by ft. This numerical process can therefore cause large round-off errors. To reduce the errors, we modify the term inside the exponential function:
Symmetrizing the operators as we do in the wave equation (5.19) based on the Fresnel approximation, we finally get
This is a wide-angle formulation, so the operator for the propagation over Az/2 in the Fresnel approximation corresponds to that in the wideangle formulation as follows:
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The calculation procedures for the beam propagation based on the wide-angle formulation are exactly the same as those for the beam propagation method based on the Fresnel approximation.
5.1.4
Analytical Boundaries
To reduce the reflections at analysis windows, we need some artificial boundary conditions. Since the simple transparent boundary condition (TBC) that is normally used in the FD-BPM (and which will be discussed in a later section) cannot be used in the FFT-BPM, some other artificial boundary conditions using complex refractive index materials or window functions have to be used to make the propagating fields decay properly near the edges of the analysis window. These artificial boundaries in the FFT-BPM usually require some experiences in optimizing the parameters to minimize the reflections.
5.1.5
Further Investigation
The free-space propagation operators for the Fresnel approximation and for the wide-angle formulation that were given in expression (5.44) are further examined in this section. According to Eqs. (5.12) and (5.24), the slowly varying envelope function (j)(x, z) in a 2D case is expressed as
where
and
Here, X is the total width of the analysis region and m is an integer. First, let us verify that the Fresnel approximation is equivalent to the para-axial approximation by using a ID case, where we reduce the
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derivative operator V^ in Eq. (5.44) to the square of a wave number in a lateral direction, kx. The free-space propagation operators for the Fresnel approximation and the wide-angle formulation are respectively
Thus, the phase term q>F for the Fresnel approximation and the phase term (pw for the wide-angle formulation are
Equation (5.4) can be used to express ft as a function of the reference index n eff . Assuming that the reference index neff is equal to the effective index, we can, as shown in Fig. 5.2, reduce ft to the z-directed component of the wave number A: of a whole wave (i.e., the propagation constant). Since k and ft are generally much larger than the jc-directed wave number kx, the approximation k ^ ft is valid. Thus, kx can be approximated as
FIGURE 5.2. Relation between k, kx, and ft. Here, 6 is the propagation angle.
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179
Substituting Eq. (5.52) into Eqs. (5.50) and Eq. (5.51), we can reduce the phase term cpF for the Fresnel approximation and the phase term cpw for the wide-angle formulation to
Comparing Eq. (5.53) with Eq. (5.54), we find that the Fresnel approximation is a good approximation for the wide-angle formulation only when the propagation angles are so small that cos2(0/2) can be considered to be nearly equal to 1. Next, we discuss the discretization width in the lateral directions. Since the free-space propagation operator (5.48) for the Fresnel approximation is purely imaginary, it satisfies the unitary condition. The free-space propagation operator (5.49) for the wide-angle formulation, on the other hand, has to satisfy
if the unitary condition is to be satisfied. That is, when
is not satisfied, the denominator of the argument of the operator (5.49) becomes complex. Thus, the free-space propagation operator (5.49) itself also becomes complex and does not satisfy the unitary condition. This causes the power of the propagating optical wave to dissipate. Assuming the maximum number of m to be M/2, we can rewrite condition (5.56) as
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where Ax = X/M is the discretization width. Since the relation &o = 27r/A0 holds for the wave number in a vacuum (A0 is the wavelength in the vacuum), Eq. (5.57) implies that the discretization width Ax in the lateral direction has to satisfy the condition
if the power of the beam is to be conserved. The condition (5.58) implies that a very small discretization cannot be used in the wide-angle FFTBPM. For 3D cases, the conditions (5.57) and (5.58) are modified to
and
where we assume that Ax = Ay.
5.2
FINITE-DIFFERENCE BEAM PROPAGATION METHOD
The FD-BPM is very powerful and has been widely used for optical waveguide design. Of the various FD-BPMs that have been developed, the one with the implicit scheme developed by Chung and Dagli [4] is stateof-the-art from the viewpoints of accuracy, numerical efficiency, and stability. Its unconditional stability is particularly advantageous not only because it allows us to use the method in actual design without danger of diversion but also because it allows us to set the propagation step relatively large. In addition, the TBC [13], which is simple and requires no special experience to use, has been developed for the FD-BPM by Hadley [5]. A wide-angle scheme using Fade approximant operators [5, 6] has also been developed by him. These contributions greatly advanced the FD-BPM and have enabled it to be used even in the design of optical waveguides made of high-contrast-index materials, such as semiconductor optical waveguides.
5.2
5.2.1
FINITE-DIFFERENCE BEAM PROPAGATION METHOD
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Wave Equation
To clarify the formulation of the FD-BPM, we must first derive the wave equation. In this section, as in Chapter 2, we assume that the structure of the optical waveguide is uniform in the y direction. The necessary wave equations can be derived from the semivectorial wave equations given in Chapter 4, but let us further our understanding by deriving them from Maxwell's equations. The component representations of Maxwell's equations
were given in Eqs. (2.5)-(2.10), where the relative permeability nr was assumed to be equal to 1. Since the structure in the y direction is assumed to be uniform, the derivatives with respect to y can be set to zero. Thus, Eqs. (2.5)-(2.10) are reduced to
A. TE Mode Figure 5.3 shows the principal field components for the TE mode. Since both Ey and Hx are the principal fields, we need to have the wave equations for both. In the TE mode, as discussed in Section 2.1.1, the x- and z-directed electric field components and the j-directed magnetic field component are zero:
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FIGURE 5.3. Principal field components for TE mode are Ey and Hx.
Substituting Eq. (5.69) into Eqs. (5.63)-(5.68), we obtain for the TE mode the equations
1. Ey Representation First, we derive the wave equation for the ydirected electric field component Ey. Substituting the x- and z-directed magnetic field components
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183
obtained from Eqs. (5.70) and (5.71) into Eq. (5.72) yields the following wave equation for the principal electric field component Ey:
where &Q = ct>2e0/^0. 2. Hx Representation Next, we derive the wave equation for the xdirected magnetic field component Hx. Differentiating Eq. (5.72) with respect to z, we get
In deriving Eq. (5.76), we have assumed the variation of the relative permittivity &r along the propagation axis to be negligibly small. That is,
This results in the approximation
Thus, it should be noted that the approximation (5.77) is implicitly used in the BPM. From a calculation of 3Eq. (5.70)/3;c + 3Eq. (5.70)/9z or the magnetic divergence eauation
we get
That is,
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Substituting this equation and Eq. (5.70) into Eq. (5.76), we can eliminate Hz in Eq. (5.76). This results in the wave equation for the principal magnetic field component Hx:
B. TM Mode Figure 5.4 shows the principal field components for the TM mode. Since both Ex and Hy are principal fields, we need to have the wave equations for both. In the TE mode, as discussed in Section 2.1.2, the x- and z-directed magnetic field components and jv-directed electric field component are zero:
Substituting Eq. (5.81) into Eqs. (5.63)-(5.68), we obtain for the TM mode the equations
FIGURE 5.4. Principal field components for TM mode are Ex and Hy.
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7. Ex Representation First, we derive the wave equation for the jt-directed electric field component Ex. Differentiating Eq. (5.82) with respect to z, we get
Substituting Eq. (5.83) into Eq. (5.85), we can eliminate Hy and obtain
From a calculation of 3Eq. (5.83)/3bc + 3Eq. (5.84)/3z or the divergence equation for the electric flux density
we get
That is,
Substituting this equation into Eq. (5.86), we get the wave equation for the principal electric field component Ex:
2. Hy Representation Next, we derive the wave equation for the ydirected magnetic field component Hy. Substituting the x- and z-directed electric field components
which are obtained from Eqs. (5.83) and (5.84), into Eq. (5.82) yields
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and therefore
Making use of the approximation
which follows from Eq. (5.77), we get the wave equation for the principal magnetic field component Hy:
5.2.2
FD-BPM Formulation
Next, let us discuss the FD-BPM formulation based on the implicit scheme developed by Chung and Dagli [4]. Since the discussions here are limited to 2D problems, the amount of memory required is not large. Thus, the equidistant discretization is used to ensure the second-order accuracy. Further improvement of accuracy has been achieved by Yamauchi et al. [7]. A. TE Mode Ev(x, y, z) is
The wave equation for the >>-directed electric field
As in the FFT-BPM, using the slowly varying envelope approximation, we divide the principal field Ey(x, y, z) propagating in the z direction into the slowly varying envelope function (jc, y, z) and the very fast oscillatory phase term exp(—jfiz) as follows:
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where
Here, kQ is the wave number in the vacuum and neff is the reference index, for which the effective index is usually used. Substituting
which is obtained from Eq. (5.94), into Eq. (5.93) and dividing both sides of the resultant equation by the exponential term exp(—jfiz), we get
or
where we used the relation sr — n2. As discussed in the section covering the FFT-BPM, Eq. (5.98) is the wide-angle formulation. When we assume that
Eq. (5.98) is reduced to the Fresnel wave equation
First, we discuss the FD expression for the Fresnel approximation. That for the wide-angle formulation will be covered in Section 5.3. When we use the discretization of the ;c and z coordinates
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where p and q are integers, the following notations are used for the wave function (f>(x, z) and the relative permittivity sr(x, z):
The next step is the discretization of the Fresnel wave equation (5.100). First, we discretize it in the x direction. The discretization number /, which corresponds to the z coordinate, will be discussed later. The first and the second terms on the right-hand side of Eq. (5.100) are expressed as
and
Substituting Eqs. (5.105) and (5.106) into Eq. (5.100), we get
Thus, the discretization of the wave equation (5.100) is
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where we used the definitions
The next step is the discretization of Eq. (5.107) with respect to z. Discretizing the left-hand side of Eq. (5.107) with respect to z, we get
It should be noted that, as shown in expression (5.111), the difference center of the left-hand side of Eq. (5.107) is the point l + \ midway between / and / + 1. The difference center of the right-hand side of Eq. (5.107) discretized with respect to z should be / + \. Thus, we modify Eq. (5.107) to
Rewriting this equation so that the terms on the left- and right-hand sides respectively contain 7 + 1 and /, we get
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Multiplying both sides of this equation by 2, we get the FD expression for the TE mode:
Although in Eq. (5.113) we use the wave number in a vacuum, kQ, for ease of understanding, it is recommended that in actual programming the coordinates (i.e., x, y, and z) be multiplied by k$ and the propagation constant ft be divided by A:0 in order to reduce the round-off errors. The resultant formulation corresponds to dividing both sides of Eq. (5.113)
by 4 B. TM Mode The wave equation (5.92) for the principal magnetic field component Hy is
As in Eq. (5.94), the principal field Hy(x,y, z) propagating in the z direction is divided into the slowly varying envelope function (*, y, z) and the very fast oscillatory phase term exp(—//?z) as follows:
Substituting the second derivative of Eq. (5.115), which corresponds to Eq. (5.96), into Eq. (5.114) and dividing the resultant equation by the exponential term exp(—//?z), we get for the TM mode the wide-angle wave equation
When we assume that
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191
we get the Fresnel wave equation
To discretize the wave equation (5.118), we express the x and z coordinates, the wave function 4>(x,z), and the relative permittivity er(x, z) as follows:
We first discretize the Fresnel wave equation (5.118) in the x direction. The first term on the right-hand side of Eq. (5.118) is discretized as follows:
where
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Thus, we get
We also get
Substituting Eqs. (5.126) and (5.130) into Eq. (5.118), we get
and therefore
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The next step is the discretization of Eq. (5.131) with respect to z. Discretizing the left-hand side of Eq. (5.131) with respect to z, we get
As with expression (5.111), since the difference center of the left-hand side of Eq. (5.131) is the point l + \ midway between / and / + 1 as shown in Eq. (5.131), the difference center of the right-hand side of Eq. (5.131) should be / + i. Using the same procedure, we used for the TE mode, we get the FD expression
5.2.3
Nonequidistant Discretization Scheme
In the above discussions, we first discretized the Fresnel wave equation in the x direction as
Then, discretizing Eq. (5.134) in the z direction, we got the FD expressior for the Fresnel wave equation for a ID nonequidistant discretization, which is shown in Fig. 5.5:
where the difference centers of both sides of Eq. (5.134) are the same in the z direction.
FIGURE 5.5. One-dimensional nonequidistant discretization.
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a. TE mode—Ey representation and Hx representation:
b. TM mode: Ex representation:
Hy representation:
5.2
5.2.4
FINITE-DIFFERENCE BEAM PROPAGATION METHOD
195
Stability Condition
Up to this stage, in the discretization of Eqs. (5.100) and (5.118) in the z direction, the difference centers were assumed to be the same (i.e., the point z + Az/2 midway between z and z + Az) for both the left- and the right-hand sides. Here, we further discuss the influence of the difference center of the right-hand side in the z direction on the stability along the beam propagation by using the procedure discussed in Ref. [11]. The Fresnel equation is
For simplicity, uniform media are assumed here. Since the reference index neff can be set to the refractive index JsTr of the medium, we can assume «2ff —Er Thus, the wave equation (5.145) can be simplified to
Introducing the difference parameter a, which determines the difference center in the z direction, we modify the wave equation (5.146) to get
Here, it should be noted that a — 0.5 has been assumed in the preceding discussions and that the scheme with a = 0.5 is called the CrankNicolson scheme. When for a plane wave the slowly varying envelope function is exoressed as
the second derivative of the wave function with respect to jc is
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Substituting Eqs. (5.148) and (5.149) into (5.147) and dividing the resultant equation by (j)(x, z), we get
Therefore
and
Since the exponential term exp(—//? Az) given by Eq. (5.150) is a propagation term, we can clarify the influence of the parameter a on the stability of the beam propagation by investigating how the absolute value of the exponential function changes when the propagation distance Az changes. The absolute value of the propagation term is expressed as
where
The absolute value of the propagation term is related to the value of a as follows: CASE a = 0.5
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Since the absolute value of the propagation term is always equal to 1, the propagating field does not diverge as the beam propagates. There is, however, the possibility of oscillation. CASE 0.5 < a < 1
Since the absolute value of the propagation term is always less than 1, the propagating field is unconditionally stable. However, the propagating field decays as the beam propagates. CASE 0 < < x < 0.5
Since the absolute value of the propagation term is always greater than 1, the propagating field grows larger and larger as the beam propagates. It will finally diverge. A difference parameter a less than 0.5 thus should not be used, and a = 0.5 is usually used in the calculation. The Fresnel wave equation with the difference parameter a is expressed as
Equation (5.156) is solved in exactly the same way that Eqs. (5.112) and (5.133), which are based on the Crank-Nicolson sheme, are solved. 5.2.5
Transparent Boundary Condition
An infinitely wide area would have no analysis boundary and there would thus be no reflections at boundaries. Because such an area cannot be assumed in actual design, however, a limited analysis window has to be
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used. In actual structures, radiated waves are reflected at the boundaries and return to the core area, where they interact with the propagating fields. This interaction disturbs the propagating fields and greatly degrades the calculation accuracy. In this section, we discuss the TBC. The TBC was developed by Hadley [5] as a way to efficiently suppress the reflections at boundaries, and it is easy to implement into computer programs. Although the TBC will be applied to 2D problems here, it can easily be extended and applied to 3D problems. As shown in Fig. 5.6, the analysis window contains nodes atp = 1 to p = M. The hypothetical nodes at p = 0 and p = M + 1 are assumed to be outside the analysis window. Following Hadley's line of thinking [5], we incorporate the influences of the nodes at p = 0 and p = M + 1 into the nodes at p — 1 and p — M. In what follows, we consider how the boundary conditions can be written into a computer program. A. Left-Hand Boundary Consider the left-hand boundary in Fig. 5.6. We incorporate the influence of the hypothetical node atp = 0 (outside the analysis window) into the node at p = 1 (inside the analysis window). The wave function for the left-traveling wave with the jc-directed wave number kx is expressed as
We denote the ;c coordinates and the fields of the nodes atp = 0, 1, 2 as #o, jc l5 x2 and as 00, 4>l, (j)2, and we assume that
The fields <j)l and (f)2 are inside the analysis window, and 0 is a hypothetical field whose influence should be incorporated into the field inside the analysis window.
FIGURE 5.6. Nodes p = 0 and p = M + 1 are outside the analysis area.
5.2
FINITE-DIFFERENCE BEAM PROPAGATION METHOD
199
Dividing Eq. (5.160) by Eq. (5.159) and Eq. (5.159) by Eq. (5.158), we get
where Ax = x2 — x{ = xl — XQ. Substituting the ratio of 2 to 0i>
which is equal to &xp(jkx Ax), into Eq. (5.162), we get
Equation (5.164) can also be derived by substituting the ^-directed wave number
which is obtained from Eq. (5.161), into
which is obtained from Eq. (5.162). It should be noted that since the wave travels leftward, the real part of the ^-directed wave number kx, Re(kx), should be negative. When it is positive, which implies reflection at the left-hand boundary, the sign should be changed from plus to minus. B. Right-Hand Boundary Consider the right-hand boundary in Fig. 5.6. We incorporate the influence of the hypothetical node at/? = M + 1 (outside the analysis window) into the node at p — M (inside the analysis window).
200
BEAM PROPAGATION METHODS
The wave function of the right-traveling wave with the x-directed wave number kx is expressed as
We denote the x coordinates and the fields of the nodes/? = M — l,M, M + 1 as xM_lt XM, xM+l and <^_1? $M, (j>M+i, and we assume that
where <j>M_\ and M are the fields inside the analysis window and A/+1 is the hypothetical field whose influence should be incorporated into the field inside the analysis window. Dividing Eq. (5.169) by Eq. (5.168) and Eq. (5.170) by Eq. (5.169), we get
where Ax = XM — XM_I = xM+l —XM. Substituting the ratio of (f)M <£M-I>
to
which is equal to &xp(jkx Ax), into Eq. (5.172), we get
Equation (5.174) can also be derived by substituting the ^-directed wave number
5.2
FINITE-DIFFERENCE BEAM PROPAGATION METHOD
201
which is obtained from Eq. (5.171), into
Similar to what we saw in the case of the left-hand boundary, since the wave travels rightward, the real part of the ^-directed wave number kx should be negative. When it is positive, which implies reflection occurs at the right-hand boundary, the sign should be changed from plus to minus. 5.2.6
Programming
We now look at how the TBC is written into the program. For the FD scheme in the propagation direction, the most widely used CrankNicolson scheme (i.e., a = 0.5) in Eq. (5.147) is used. The problem is to obtain the unknown field (/>/+1 at z + Az by using the known field 4>l, where superscripts / and / + 1 respectively correspond to z and z + Az. The following equation has to be solved:
The unknown fields to be obtained at z + Az are $^\, ^+1, and <^+\ and the known fields at z are <^,_15 lp, and 4>lp+l for/? = 1 and M. Assuming the unknown coefficients A(p), B(p), and C(p) and the known value D(p) to be
we simplify Eq. (5.177) to
202
BEAM PROPAGATION METHODS
In the following, we will discuss Eq. (5.182) with respect to p, which represents the lateral position of the node. A. Lett-Hand Boundary (p=l) The field 4>l of the node on the lefthand boundary is influenced by the field 0 of the hypothetical node outside the analysis window. As shown in Eq. (5.164), the field (f>Q of the hypothetical node is expressed as
where
Here, since the parameter JL is determined by the known fields at z (i.e., /), yL is known. Thus, Eq. (5.182) is reduced to
where
B. Right-Hand Boundary (p = M) The field $M of the node on the right-hand boundary is influenced by the field M+1 of the hypothetical node outside the analysis window. As shown in Eq. (5.174), the field 4>M+\ of the hypothetical node is expressed as
where
5.2
FINITE-DIFFERENCE BEAM PROPAGATION METHOD
203
Here, since the parameter yR is determined by the known fields at z (i.e., /), yL is known. Thus, Eq. (5.182) is reduced to
where
Summarizing the above discussions, we get the algebraic equations
Since Eq. (5.195) is a tridiagonal matrix equation, we can obtain the unknown wave fields 0i + 1 ,..., (f)1^1 by using the Thomas method [14].
204
BEAM PROPAGATION METHODS
5.3 WIDE-ANGLE ANALYSIS USING FADE APPROXIMANT OPERATORS Up to this stage, the discussions have been based on the Fresnel equation (i.e., the para-axial wave equation). Here, we discuss the wide-angle beam propagation method based on Fade approximant operators [5] and discuss the multistep method [6], both of which were developed by Hadley.
5.3.1
Pade Approximant Operators
When the second derivative with respect to z is not neglected, the wave equation is
where for the TE mode
and for the TM mode
Solving Eq. (5.196) formally, we get
and therefore
5.3 WIDE-ANGLE ANALYSIS USING FADE APPROXIMANT OPERATORS
205
When the derivative with respect to z is neglected, Eq. (5.199) is reduced to the Fresnel equation. Here, we regard the derivative with respect to z in Eq. (5.199) as the recurrence formula
Next, we will specify the explicit expressions for the various orders of the recurrence formula. First, we define the starting equation
The explicit expressions for the corresponding wide-angle (WA) orders are shown below. 1. WA-Oth order (Fresnel approximation):
2. WA-lst order:
3. WA-2nd order:
206
BEAM PROPAGATION METHODS
4. WA-3rd order:
5. WA-4th order:
6. WA-5th order:
5.3 WIDE-ANGLE ANALYSIS USING PADE APPROXIMANT OPERATORS
207
7. WA-6th order:
8. WA-7th order:
Thus, the recurrence formula (5.200) can be reduced to an expression that includes only the operator P:
208
BEAM PROPAGATION METHODS
where N and D are both polynomials of the operator P. The wide-angle orders correspond to the orders for the Fade orders as follows:
Differentiating Eq. (5.210) based on the Crank-Nicolson scheme, we get
where the right-hand side was averaged by / and / + 1 so that the difference center of the left-hand side coincides with that of the righthand side. From Eq. (5.211), we get
Therefore
and
which can be rewritten as
5.3 WIDE-ANGLE ANALYSIS USING PADE APPROXIMANT OPERATORS
209
Since, as shown in Eqs. (5.202)-(5.209), the coefficients of the polynomials D and N in Eq. (5.213) are real, D and N themselves are real. Thus, Eq. (5.213) can be written as
In the following, we show the coefficients £z for the wide-angle orders WA-0 to WA-7. 1. WA-Oth order [Pade(l,0): Fresnel approximation]: From Eq. (5.202), we get
and therefore
Thus, we get
2. WA-lst order [Pade(l,l)]: From Eq. (5.203), we get
and therefore
210
BEAM PROPAGATION METHODS
3. WA-2nd order [Pade(2,l)]: From Eq. (5.204), we get
and therefore
Thus, we get
4. WA-3rd order [Pade(2,2)]: From Eq. (5.205), we get
and therefore
Thus, we get
5.3
WIDE-ANGLE ANALYSIS USING PADE APPROXIMANT OPERATORS
5. WA-4th order [Pade(3,2)]: From Eq. (5.206), we get
and therefore
Thus, we get
6. WA-5th order [Pade(3,3)]: From Eq. (5.207), we get
and therefore
211
212
BEAM PROPAGATION METHODS
7. WA-6th order [Pade(4,3)]: From Eq. (5.208), we get
and therefore
Thus, we get
5.3 WIDE-ANGLE ANALYSIS USING FADE APPROXIMANT OPERATORS
213
8. WA-7th order [Pade(4,4)]: From Eq. (5.209), we get
and therefore
Thus, we get
Since through the above discussions the explicit expressions for D —j(Az/2)N are obtained, those for D +j(Az/2)N can also be obtained. Both sides of Eq. (5.212) are clarified and the unknown field 0/+1 can be calculated.
214
BEAM PROPAGATION METHODS
Since the wide-angle formulations written in Eqs. (5.204)-(5.209) or Eqs. (5.219)-(5.230) include the powers of the operator P higher than 2, the column width of nonzero matrix elements is greater than 3. Thus, the numerically efficient Thomas method cannot be used to solve the final algebraic matrix equation. In this section, we discuss the multistep method, which was developed by Hadley [6] in order to solve this problem. Consider Eq. (5.214). The numerator of the factor on the right-hand side of Eq. (5.214) is obtained as shown in Eqs. (5.215)-(5.230). The denominator can also be obtained, since it is simply the complex conjugate of the numerator. Since £0 is equal to 1, the numerator of the term on the right-hand side of Eq. (5.214) can be factorized as
where the coefficients al,a2, • •-,an can be obtained by solving the algebraic equation
The denominator of the term on the right-hand side of Eq. (5.214) can be obtained by using the complex conjugates of the coefficients a l s a2,...,an:
Thus, the unknown field /+1 at z + Az is related to the known field (f)1 at z as follows:
Next, we discuss how to solve Eq. (5.234). First, we rewrite it as
5.3
WIDE-ANGLE ANALYSIS USING PADE APPROXIMANT OPERATORS
215
Then, defining the field /+1/n as
we rewrite Eq. (5.235) as
Since /+1/n , we rewrite Eq. (5.236) as
Then, defining the field 0/+2/w as
we rewrite Eq. (5.238) as
Since cf)l+l/n is known, we can obtain cj)l+2/n by solving Eq. (5.240). Repeating this procedure, we finally get the unknown field /+1 at z + Az by solving
That is, the known field (f)l+l can be obtained from the known field 4>l by successively solving
when i = 1, 2 , . . . , « . The advantage of the multistep method is that the matrix equation to be solved in each step is the same size as the Fresnel equation and for 2D problems is tridiagonal. Thus, the calculation procedure is very easy. The method can be easily extended to 3D problems, and it has also been used in the wide-angle FE-BPM [12].
216
5.4
BEAM PROPAGATION METHODS
THREE-DIMENSIONAL SEMIVECTORIAL ANALYSIS
The preceding discussions were limited to the 2D BPM, which assumes the ID cross-sectional structure in the lateral direction. When, however, the propagating field is widely spread in the 2D cross section, the 3D beam propagation method is required. Since the formulation for the 2D BPM in Section 5.2.2 can be straightforwardly extended to the 3D BPM, a much more numerically efficient 3D BPM formulation based on the alternate-direction implicit (ADI) method [8-10] will be shown here. In the ADI-BPM, the calculation for one step, z ->• z + Az, is divided into two steps, z —»• z + Az/2 and z + Az/2 —>• z + Az, and the two steps are solved successively in the x and y directions. Since solving a 3D problem can be reduced to solving a 2D problem twice by using the ADI method, instead of a large matrix equation we have only to solve tridiagonal matrix equations twice. Thus, high numerical efficiency is attained especially for large-size waveguides, such as spotsize-converterintegrated structures [15-17]. In this section, the semivectorial formulation is used to analyze large-index-difference waveguides and to treat the polarization. Nonuniform discretization is also assumed for versatility of analysis. Neglecting the terms for the interaction between polarizations in the vectorial wave equations (4.10) and (4.19) in Section 4.2, we get the semivectorial wave equation
Here, P\l/ and \l/ for the quasi-TE mode are obtained from Eqs. (4.17) and (4.30) as
5.4 THREE-DIMENSIONAL SEMIVECTORIAL ANALYSIS
217
And P\l/ and \j/ for the quasi-TM mode are obtained from Eqs. (4.18) and (4.29) as
Using the slowly varying envelope approximation—in this case, that the wave function \j/(x, y, z) propagating in the z direction can be separated into a slowly varying envelope function (j)(x,y,z) and a very fast oscillating phase term exp(—jfiz)—we get
where k0, weff, and ft (= «eff&o) are respectively the wave number in a vacuum, the reference index, and the propagation constant. Assuming the Fresnel approximation
we reduce Eq. (5.243) to the Fresnel wave equation r*
/
It should be noted that PI/S in Eq. (5.250) differs by -f from that in Eqs. (5.243)-(5.247) and for the quasi-TE mode is expressed as
in the electric field representation and as
218
BEAM PROPAGATION METHODS
in the magnetic field representation. For the quasi-TM mode it is expressed as
in the electric field representation and as
in the magnetic field representation. The nonequidistant discretization mesh shown in Fig. 4.3 for the 2D cross-sectional FDM is also used in the lateral directions for the 3D FDBPM. Here, the subscripts for the lateral positions x and y are respectively p and q and the superscript for position z in the propagation direction is /. Thus, using Eqs. (4.51)-(4.59), we can write the fields, the discretization widths, and the relative permittivity as follows:
First, we discuss the discretization with respect to x and y. Discretizing Eqs. (5.251M5.253), by deduction from Eqs. (4.99), (4.123), (4.133), and (4.143), we get
where ft = &oneff • Thus, discretizing only the right-hand side of Eq. (5.250) with respect to x and y, we can reduce Eq. (5.250) to
5.4 THREE-DIMENSIONAL SEMIVECTORIAL ANALYSIS
219
This equation can be rewritten as
Now, we move to the discretization of the derivative with respect to z on the left-hand side of Eq. (5.257). A sensitive problem in discretization is the difference centers of the right-hand side and the left-hand side of the equation in the z direction. In the ADI-BPM, the calculation for the step z —»• z + Az is divided into two steps, z —> z + Az/2 and z + Az/2 —>• z + Az. In the following, we describe the explicit calculation procedure. 5.4.1
First Step: z -> z + Az/2(/ -»/ + £)
The derivative with respect to x on the right-hand side of Eq. (5.257) is written by the implicit FD expression using the unknown fields at / + \ as
The derivative with respect to y, on the other hand, is written by the explicit FD expression with the known fields at / as
The remaining part on the right-hand side is expressed by using an average of / and / + \ as
where ej.+1/2 = (ej. + 4+1)/2. With respect to the left-hand side of Eq. (5.257), we get
220
BEAM PROPAGATION METHODS
Thus, from expressions (5.258)-(5.261), we get
So that the terms on the left- and right-hand sides respectively contain l + \ and /, we rewrite Eq. (5.262) as
5.4.2
Second Step: z + Az/2 -> z + Az(7 + £ - > / + ! )
The derivative with respect to y on the right-hand side of Eq. (5.257) is written by the implicit FD expression using the unknown fields at / + 1 as
The derivative with respect to x, on the other hand, is written by the explicit FD expression using the known fields at / + \ as
The remaining part in the right-hand side is expressed by using an average of / +1 and / + 1 as
5.4 THREE-DIMENSIONAL SEMIVECTORIAL ANALYSIS
221
With respect to the left-hand side of Eq. (5.257), we get
Thus, from Eqs. (5.264)-(5.267), we get
So that the terms on the left- and right-hand sides respectively contain / + 1 and / +1, we rewrite Eq. (5.268) as
As discussed above, since the actual calculation in the two steps of the ADI-BPM is the 2D BPM, it is very numerically efficient. The mode mismatch (obtained from the overlap integral) between the propagating field calculated by the semivectorial ADI-BPM and the initial eigenfield calculated by the scalar FEM is shown in Fig. 5.7 as a function of the propagation distance. Since the propagating field and the initial field are obtained by the semivectorial ADI-BPM and the scalar FEM, the mode mismatch shown in Fig. 5.7 corresponds to the difference between the eigenfield shapes of the FD scheme and the FE scheme. As shown in this figure, the difference between the eigenfields obtained by the FDM and the FEM is very small even though the concepts of the two methods are different.
222
BEAM PROPAGATION METHODS
FIGURE 5.7. Eigenfield mismatch between the FEM and FD-BPM. 5.5
THREE-DIMENSIONAL FULLY VECTORIAL ANALYSIS
Since a full discussion of the 3D fully vectorial beam propagation method is beyond the scope of this book, we roughly cover the formulation, emphasizing the relations between the wave equations. For details, refer to Ref. [11]. 5.5.1
Wave Equations
As shown in Eqs. (4.17) and (4.18), the 3D vectorial wave equations for the electric field representation are for the jc component
and for the y component
As shown in Eqs. (4.29) and (4.30), the 3D vectorial wave equations for the magnetic field representation are for the x component
5.5 THREE-DIMENSIONAL FULLY VECTORIAL ANALYSIS
223
and for the y component
First, we discuss the electric field representation. Using the SVEA—in this case the wave functions Ex(x, y, z) and Ey(x, y, z) propagating in the z direction can be separated into the slowly varying envelope functions Ex(x, y, z) and Ey(x,y,z) and the vary fast oscillating phase term exp(—jfiz)—we get
Substituting the second derivatives with respect to z,
into Eqs. (5.274) and (5.275) and dividing the results by the phase term exp(—//?z), we get the wide-angle equations for the electric field representation:
The matrix expression for Eqs. (5.278) and (5.279) is
224
BEAM PROPAGATION METHODS
where
using tne svbAs
we get the wide-angle wave equation for the magnetic field representation:
where
Neglecting the interaction terms P^ and Pyx in Eqs. (5.280) and (5.287), we get the semivectorial wave equations. Neglecting the second terms on
5.5 THREE-DIMENSIONAL FULLY VECTORIAL ANALYSIS
225
the left-hand sides of Eqs. (5.280) and (5.287), we get the vectorial wave equations based on the Fresnel approximation.
5.5.2
Finite-Difference Expressions
Here, we discuss the fully vectorial FD-BPM using as an example the vectorial wave equations (5.287)-(5.291) for the magnetic field representation. For simplicity, the equidistant discretizations Ax and Ay are assumed in the x and y directions. Since P^ is an operator corresponding to the semivectorial analysis for Hx,
where, from Eqs. (4.143)-(4.149), we get
226
BEAM PROPAGATION METHODS
On the other hand,
where, from Eqs (4.123) to (4.129), we get
Although the derivation is not shown here, the FD expressions for the interaction terms PxyHy and PyxHx (for the interaction between
PROBLEMS
227
Hx and Hy) are
The final FD beam propagation equations to be solved can be obtained by applying the FD procedure to the derivative with respect to the z coordinate.
PROBLEMS 1. Using a plane wave, evaluate the difference error for the FD approximation used in the FD-BPM.
ANSWER When we use the center difference scheme, the second derivative of the wave function (f)(x) with respect to x is
228
BEAM PROPAGATION METHODS
where Ax is the discretization width and i is the number of the node at the difference center. If we assume that wave function c/)(x) is a plane wave
the left- and right-hand sides of Eq. (5.1) can be respectively rewritten as
and
Using these two factors, we get the relative error s:
When the discretization width Ax is very small, sin^ Ax/2) can be approximated as kx Ax/2. Thus, the relative error e becomes zero as Ax^ 0. 2. Figure P5.1 shows a simple example of an analysis region for a ID FDBPM. Nodes 1—4 are inside the analysis window, and nodes 0-5 are outside the window. Show the form of the matrix equation for this example.
PROBLEMS
229
FIGURE P5.1. Simple example of a ID FD-BPM. ANSWER
where B'(l) and B'(4) include the boundary conditions due to nodes 0 and 5. For explicit forms of the coefficients on the left-hand side and of D(l) to £>(4) on the right-hand side, see Eqs. (5.178H5.181) and Eqs. (5.186) and (5.193). 3. The 3D semivectorial analysis shown in Section 5.3 was based on the ADI method. Discuss the procedure for a 3D analysis not using the ADI method. ANSWER
The starting equation is Eq. (5.256). Using a procedure similar to the one specified in Eqs. (5.131)-(5.133), we get
where the superscripts / + 1 and / respectively correspond to the unknown and the known quantities.
230
BEAM PROPAGATION METHODS
Thus, we get the 3D FD-BPM expression
REFERENCES [1] M. D. Feit and J. A. Freck, Jr., "Light propagation in graded-index optical fibers," Appl Opt., vol. 17, pp. 3990-3998, 1978. [2] L. Thylen, "The beam propagation method: An analysis of its applicability," Opt. Quantum Electron, vol. 15, pp. 433^39, 1983. [3] J. Yamauchi, J. Shibayama, and H. Nakano, "Beam propagation method using Fade approximant operators," Trans. IEICE Jpn., vol. J77-C-I, pp. 490^94, 1994. [4] Y. Chung and N. Dagli, "Assessment of finite difference beam propagation," IEEE J. Quantum Electron., vol. 26, pp. 1335-1339, 1990. [5] G. R. Hadley, "Wide-angle beam propagation using Fade approximant operators," Opt. Lett., vol. 17, pp. 1426-1428, 1992. [6] G. R. Hadley, "A multistep method for wide angle beam propagation," Integrated Photon. Res., vol. ITu 15-1, pp. 387-391, 1993. [7] J. Yamauchi, J. Shibayama, and H. Nakano, "Modified finite-difference beam propagation method on the generalized Douglas scheme for variable coefficients," IEEE Photon. Techno!. Lett., vol. 7, pp. 661-663, 1995. [8] J. Yamauchi, T. Ando, and H. Nakano, "Beam-propagation analysis of optical fibres by alternating direction implicit method," Electron. Lett., vol. 27, pp. 1663-1665, 1991. [9] J. Yamauchi, T. Ando, and H. Nakano, "Propagating beam analysis by alternating-direction implicit finite-difference method," Trans. IEICE Jpn., vol. J75-C-I, pp. 148-154, 1992 (in Japanese). [10] P. L. Liu and B. J. Li, "Study of form birefringence in waveguide devices using the semivectorial beam propagation method," IEEE Photon. Technol Lett., vol. 3, pp. 913-915, 1991.
REFERENCES
231
[11] W. P. Huang and C. L. Xu, "Simulation of three-dimensional optical waveguides by a full-vector beam propagation method," IEEE J. Quantum Electron., vol. 29, pp. 2639-2649, 1993. [12] M. Koshiba and Y. Tsuji, "A wide-angle finite element beam propagation method," IEEE Photon. Technol. Lett., vol. 8, pp. 1208-1210, 1996. [13] G. R. Hadley, "Transparent boundary condition for beam propagation," Opt. Lett., vol. 16, pp. 624-626, 1992. [14] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes, Cambridge University Press, New York, 1992. [15] K. Kawano, M. Kohtoku, M. Wada, H. Okamoto, Y. Itaya, and M. Naganuma, "Design of spotsize-converter-integrated laser diode (SS-LD) with a lateral taper, thin-film core and ridge in the 1.3 urn wavelength region based on the 3-D BPM," IEEE J. Select. Top. Quantum Electron., vol. 2, pp. 348-354, 1996. [16] K. Kawano, M. Kohtoku, H. Okamoto, Y. Itaya, and M. Naganuma, "Coupling and conversion characteristics of spot-size converter integrated laser diodes," IEEE J. Select. Top. Quantum Electron., vol. 3, pp. 13511360, 1997. [17] K. Kawano, M. Kohtoku, N. Yoshimoto, S. Sekine, and Y. Noguchi, "2 x 2 InGaAlAs/InAlAs multiple quantum well (MQW) directional coupler waveguide switch modules integrated with spot-size converters," Electron. Lett., vol. 30, pp. 353-354, 1994.
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CHAPTER 6
FINITE-DIFFERENCE TIME-DOMAIN METHOD
In the preceding chapters, steady-state wave equations were solved in which the derivative with respect to the time t (i.e., 3/3/) was replaced by y co. In this chapter, we discuss the finite-difference time-domain method (FD-TDM), which was developed by Yee [1] and which directly solves time-dependent Maxwell equations. The FD-TDM was originally proposed for electromagnetic waves with long wavelengths, such as microwaves, because the spatial discretization it requires is small (^-^ of the wavelength). As the FD-TDM is an explicit scheme, the time step in the calculation is defined by the spatial discretization width. Thus, the time step in the optical waveguide analysis is extremely short when wavelengths are of micrometer order. The amount of required memory is enormous for 3D structures, but the method is readly applicable to 2D structures. Finite-difference TDM CAD software suitable for microwave wavelengths as well as optical wavelengths is available on the market.
6.1
DISCRETIZATION OF ELECTROMAGNETIC WAVES
The 3D formulation is shown here because it is more versatile than the 2D formulation, which can be easily obtained from the 3D formulation. 233
234
FINITE-DIFFERENCE TIME-DOMAIN METHOD
The time-dependent Maxwell equations are
and using Eqs. (2.5)-(2.10), we can write their component representations as follows:
When we assume that Ax, Ay, and Az are spatial discretizations and that A? is a time step, the function F(JC, y, z, t) is discretized as
Figure 6.1 shows what we call the Yee lattice [1]. Using a to represent a spatial coordinate such as jc, y, and z, we define
Difference centers play important roles when Eqs. (6.3)-(6.8) are discretized, and here we investigate Eq. (6.3) as an example. Since the spatial difference centers of the left-hand side of the equation are the same
6.1
DISCRETIZATION OF ELECTROMAGNETIC WAVES
235
FIGURE 6.1. Yee lattice.
as those for Hx, the spatial difference centers in the x, y, and z directions are respectively found to be jc — i Ax, y = (j + ^ Ay, and z = (k + £) Az. Since the time difference center of the right-hand side of the equation is the same as that for the electric fields Ey and Ez, we can write t = n Af. Here, i,j, k, and n are integers. In a similar manner, we can also obtain the following difference centers of the spatial coordinates and the time for Eqs. (6.3H6-8).
236
FINITE-DIFFERENCE TIME-DOMAIN METHOD
The 3D finite-difference time-domain expressions for Eqs. (6.3)-(6.8) can be obtained by discretizing them on the basis of the difference centers (6.12)-(6.17). Again, we investigate Eq. (6.3) as an example. The difference centers x and t are both integers and the difference centers y and z are both half-integers. Thus, for the left-hand side of Eq. (6.3), we get
For the right-hand side, we get
Using expressions (6.18) and (6.19), we get the following finite-difference time-domain expression for Eq. (6.3):
Through the same procedure, we get the following finite-difference time-domain expressions for the y and z components of the magnetic fields:
6.1
DISCRETIZATION OF ELECTROMAGNETIC WAVES
237
and
Next, we discretize Eq. (6.6). According to expression (6.15), the difference centers of x and t are both half-integers and the difference centers y and z are both integers. Thus, for the left-hand side of Eq. (6.6), we get
For the right-hand side, we get
Using expressions (6.23) and (6.24), we get the following finite-difference time-domain expression for Eq. (6.6):
238
FINITE-DIFFERENCE TIME-DOMAIN METHOD
Through the same procedure, we get the following finite-difference time-domain expressions for thejy and z components of the electric fields:
and
Magnetic fields H"+ ' with the half-integer time step (n 4- 1 /2) At are calculated first from Eqs. (6.20)-(6.22) by using the electric fields with the integer time step n At. Then those fields are used to calculate the electric fields E"+l with the integer time step (n + 1) At by using Eqs. (6.25)(6.27). Repeating these two steps, we can calculate the time evolution of the electric and magnetic fields directly. It should be noted that the relative permittivity at the interface between two media is approximated better by using (erl + er2)/2 than by using only Erl or er2.
6.2
STABILITY CONDITION
239
6.2 STABILITY CONDITION In an explicit scheme such as the FD-TDM, the time step Af in the calculation is restricted by the spatial discretization. For simplicity in discussing the stability condition here, we will use the ID scalar Helmholtz equation
where (/> is a ID wave function that designates the time-dependent field. Using fix as the ^-directed propagation constant, we express this wave function as
where £ = exp(a A/). Thus, if the field is to be stable, £ has to satisfy the condition
Substituting Eq. (6.29) into (6.28), we get
Dividing this equation by Gxp(jfixp
Ax)£", we reduce it to
240
FINITE-DIFFERENCE TIME-DOMAIN METHOD
Dividing Eq. (6.31) by £/^0/(Af)2£ and considering that the first term of Eq. (6.31) can be rewritten as
we get
and therefore
where the parameter A is defined by
The roots of Eq. (6.32) are
Because |£| < 1 and 0 < sin2 6, we get the relation
We can thus specify the stability condition in terms of A: Case A < — 1. Since, according to Eq. (6.35), 1 < |£2|, the field is unstable.
6.3
ABSORBING BOUNDARY CONDITIONS
241
Case — 1 < A < 1. Since ^ and £2 can be expressed as
their absolute values can be expressed as
Thus, the field is stable when
Relation (6.40) can be interpreted as imposing the following restriction on the time step (see Problem 1):
where Ax; is the spatial discretization width and A^ is the time step and where c0, £r, and v = c0/^/s^ are respectively the velocity of the light in a vacuum, the relative permittivity of the medium, and the velocity of the light in the medium. Equation (6.41) is for a ID structure, and the corresponding restriction for a 3D structure is
6.3
ABSORBING BOUNDARY CONDITIONS
Since the FD-TDM, like the BPM in Chapter 5, has finite analysis windows, an artificial boundary condition suppressing reflections at the analysis windows is required. Mur's absorbing boundary condition (ABC) [2] is often used for this purpose, though Berenger's perfectly matched layer (PML) scheme [3] has also come into use recently. The PML scheme suppresses reflections better than Mur's ABC does, but Mur's condition is easier to use. Here, we discuss Mur's first-order ABC.
242
FINITE-DIFFERENCE TIME-DOMAIN METHOD
As shown in Fig. 6.2, the analysis window is defined in ranges of (0, Lx) in the x direction, (0, Ly) in the y direction, and (0, Lz) in the z direction. 1. x = 0: Ey and Ez. The electric fields Ey and Ez are on the boundary jc = 0. The wave function W for the left-traveling wave incident perpendicular to the boundary is
where vx is the velocity of the wave. Thus, the derivatives of the wave function with respect to x and t are
Substituting Eq. (6.45) into (6.44), we get
FIGURE 6.2. Analysis region.
6.3 ABSORBING BOUNDARY CONDITIONS
243
and therefore
Thus, the wave equation for Mur's first-order ABC is
Next, we discretize the wave equation (6.46) for the case that the wave function is that for the y-directed electric field Ey. Assuming that the node number of the node on the boundary is 0, we discretize the derivative of the electric field Ey with respect to the coordinate x as
where the time average between n and n + 1 was taken on the right-hand. On the other hand, we discretize the derivative of the electric field Ey with respect to time t as
where the spatial average between i = 0 and i = 1 was taken. Substituting Eqs. (6.47) and (6.48) into Eq. (6.46), we can derive the following finitedifference time-domain expression for the electric field Ey:
244
FINITE-DIFFERENCE TIME-DOMAIN METHOD
We can similarly derive the finite-difference time-domain expression for the electric field Ez:
2. x = Lx: Ey and Ez. The electric fields Ey and Ez are on the boundary x = Lx. The wave function W for the right-traveling wave incident perpendicular to the boundary is
where vx is a velocity of the wave. Thus, the derivatives of the wave function with respect to x and t are
Substituting Eq. (6.53) into (6.52), we get the wave equation
Next, we discretize the wave equation (6.46) for the case that the wave function is the jy-directed electric field Ey. Assuming that the node number of the node on the boundary is Nx, we discretize the derivative of the electric field Ey with respect to x as
PROBLEMS
245
where the time average between n and n + 1 was taken on the right-hand side. On the other hand, we discretize the derivative of the electric field Ey with respect to time t as
where the spatial average between / = A^ and Nx — \ was taken. Substituting Eqs. (6.55) and (6.56) into Eq. (6.54), we can derive the finitedifference time-domain expression for the electric field Ey\
We can similarly derive the finite-difference time-domain expression for the electric field Ez:
For the ABCs on y — 0 and y = Ly and on z = 0 and z = Lz, see Problem 2.
PROBLEMS 1. Derive the restriction on the time step A? specified in Eq. (6.41) by using the stability condition — 1 < A < 1 in relation (6.40).
246
FINITE-DIFFERENCE TIME-DOMAIN METHOD
ANSWER Using Eq. (6.33), we can rewrite the stability condition in relation (6.40) as
Since the relation between the center term and right-hand term is always satisfied, we have to consider only the left-hand term and center term:
Multiplying both sides by — 1 and considering the case in which the righthand side reaches its maximum, when sin2(-) = 1, we can rewrite the above relation as
Thus, we get
and therefore
And this relation can be rewritten as the restriction on the time step A/:
2. Derive the ABC fields for y = 0 and y = Ly and z = 0 and z = Lz.
PROBLEMS
247
ANSWER a. y = 0. The wave equation on this boundary is
The finite-difference time-domain expressions for the electric fields Ex and Ez on this boundary are
b. y — Ly. The wave equation on this boundary is
The finite-difference time-domain expressions for the electric fields Ex and Ez on this boundary are
248
FINITE-DIFFERENCE TIME-DOMAIN METHOD
c. z = 0. The wave equation on this boundary is
The finite-difference time-domain expressions for the electric fields Ex and Ey on this boundary are
d. z = Lz. The wave equation on this boundary is
The finite-difference time-domain expressions for the electric fields Ex and Ey on this boundary are
REFERENCES
249
REFERENCES [1] K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propagat., vol. AP-14, pp. 302-307, 1966. [2] G. Mur, "Absorbing boundary conditions for the finite-difference timedomain approximation of the time domain electromagnetic field equations," IEEE Trans. Electromagn. Compat., vol. EMC-23, pp. 377-382, 1981. [3] J.-P. Berenger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Computat. Phys., vol. 114, pp. 185-220, 1994.
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CHAPTER 7
SCHRODINGER EQUATION
In this chapter, we will investigate ways to solve the Schrodinger equation, which must be solved when quantum wells for semiconductor optical waveguide devices are designed [1, 2]. The time-dependent Schrodinger equation resembles the Fresnel wave equation, and the time-independent Schrodinger equation resembles the wave equation for the cross-sectional analysis. Thus, the analysis techniques given earlier for optical waveguides can be applied to the analysis of these Schrodinger equations. Here, the analysis of the time-dependent Schrodinger equation will be based on the 2D FD-BPM and the analysis of the time-independent Schrodinger equation will be based on the ID FDM and the ID FEM.
7.1
TIME-DEPENDENT STATE
Let us solve the time-dependent Schrodinger equation by using the 2D FD-BPM. The only major difference between the time-dependent Schrodinger equation and the BPM wave equation based on the Fresnel approximation is that the derivative in the Schrodinger equation is with respect to time, whereas the derivative in the BPM wave equation is with respect to position. 251
252
SCHRODINGER EQUATION
Under the effective mass approximation, the time-dependent Schrodinger equation is expressed as
where U(x), m(x), and h are respectively an arbitrary potential shown in Fig. 7.1, an effective mass, and the Plank constant [3]. First, the time t and the space x are discretized. The nonequidistant discretization shown in Fig. 5.5 is assumed, and the wave function \j/(x, t), the potential U(x), and the effective mass m(x) are expressed as
The 2D Fresnel wave equation for the TM mode of an optical waveguide that was derived in Chapter 5 [Eq. (5.118)] is
Comparing this with the time-dependent Schrodinger equation (7.1), we find the following correspondences:
FIGURE 7.1. Potential distribution for a semiconductor quantum well.
7.2
FINITE-DIFFERENCE ANALYSIS OF TIME-INDEPENDENT STATE
253
Thus, from Eq. (5.126) and Eqs. (5.142)-(5.144), we can get the finitedifference expression for the right-hand side of correspondence (7.7):
where
Considering the relations shown in correspondences (7.6)-(7.9) and in Eq. (5.135), we get the following finite-difference expression for the timedependent Schrodinger equation:
It is easily understood that this equation can be solved in the same way that the 2D FD-BPM is solved and that the transparent boundary condition can also be used with this equation.
7.2 FINITE-DIFFERENCE ANALYSIS OF TIME-INDEPENDENT STATE The wave function \l/(x, f) that satisfies the time-dependent Schrodinger equation (7.1) is divided into the space-dependent term and the time-
254
SCHRODINGER EQUATION
dependent term as follows:
where E is the eigenenergy to be calculated and corresponds to the effective index in the optical waveguide problems. Substituting Eq. (7.15) into Eq. (7.1) and dividing the resultant equation by exp(—jEt/h\ we get
This is the time-independent Schrodinger equation. Discretizing it by using Eqs. (7.10)-(7.13), we get the following finite-difference expression:
and therefore
The remaining task is to construct a matrix equation for Eq. (7.17).
7.3 FINITE-ELEMENT ANALYSIS OF TIME-INDEPENDENT STATE 7.3.1
Eigenvalue Equation
To solve the time-independent Schrodinger equation, we first derive the eigenvalue equation based on the FEM [4, 5]. The following normalizations are introduced:
7.3
FINITE-ELEMENT ANALYSIS OF TIME-INDEPENDENT STATE
255
where E™ = h2n2/(2mQW2), m0, and W are respectively the electron energy of the ground state in an infinitely deep well, the static mass of an electron, and the width of a quantum well. When we substitute these normalized parameters into Eq. (7.16), we get the normalized Schrodinger equation
Next, we use the Galerkin method to solve this equation. For simplicity, the overbar, denoting normalized quantities in Eqs. (7.18) to (7.22), is omitted. Figure 7.2 shows nodes used in the ID FEM. Using shape functions, we expand the wave function (*) for an element e:
Substituting Eq. (7.23) into Eq. (7.22), we get
Multiplying the left-hand side of this equation by the shape function and integrating the resultant equation in element e, we get
Applying the partial integration to the first term of Eq. (7.24) and denoting the node numbers of the left- and right-hand nodes in element e as / and i+l , we rewrite Eq. (7.24) as
FIGURE 7.2. Nodes in the ID finite-element method.
256
SCHRODINGER EQUATION
Furthermore, assuming that both the effective mass m and the potential U(x) (— Ue) are constant in the element, we can reduce this equation to
Next, we sum Eq. (7.25) for all elements. The first term of Eq. (7.25) becomes
where M is the total number of nodes. It should be noted that the following continuity conditions for the wave function and its derivative are assumed at adjacent elements e and e + 1:
Thus, after summing Eq. (7.25) for all elements, we get
7.3
FINITE-ELEMENT ANALYSIS OF TIME-INDEPENDENT STATE
257
This equation can be reduced to
where
Assuming the Dirichlet condition or the Neumann condition—that is, assuming
or
at the leftmost node 1 and the rightmost node M—we can obtain from Eq. (7.28) the simple eigenvalue equation
To solve this equation, we have to transform Eqs. (7.28) and (7.35) into eigenvalue matrix equations. To this end, in the following, the first- and second-order shape functions will be obtained and the explicit expressions for the matrixes will be shown. 7.3.2
Matrix Elements
A. First-Order Line Element The matrixes for the eigenvalue equation will be calculated by using the first-order line element. Figure 7.3 shows the first-order line element. The node numbers / and j and the coordinates Xj and Xj are assumed to correspond to the local coordinates 1
258
SCHRODINGER EQUATION
and 2. An arbitrary coordinate x in element e is defined using the parameter £, which takes a value between 0 and 1:
where Le is the length of the element (Xj — xt). Since the first-order line element has two nodes, the wave function (f)e(x) in element e is expanded as
by using the shape function [N]T. The shape functions NI and N2 for the line elements are expressed by the linear functions
and the conditions that must be met by the shape functions are
Thus, the following shape functions can be obtained for the first-order line element:
Next, we calculate the matrix elements shown in Eqs. (7.29)-(7.32).
FIGURE 7.3. First-order line element.
7.3
FINITE-ELEMENT ANALYSIS OF TIME-INDEPENDENT STATE
and therefore,
Since the relations
hold, we get
Thus, we get
2. fe [Ne][Ne]T dx
Through a similar procedure, we get
259
260
SCHRODINGER EQUATION
Since Eqs. (7.47) and (7.48) can be used to construct the matrixes [jP], [Q], and [R], the eigenenergy can be calculated from the eigenvalue matrix equation (7.35). B. Second-Order Line Element Next, we discuss the second-order line element, which is more accurate than the first-order line element. Figure 7.4 shows the second-order line element. The node numbers i, j, and k and the coordinates xt, Xj, and xk are assumed to correspond to the local coordinates 1,2, and 3. An arbitrary coordinate x in element e is defined using the parameter ^, which takes a value between — 1 and 1:
Since the second-order line element has three nodes, the wave function (f)e(x) in element e is expanded as
by using the shape function [N] . The shape functions N{, N2, and N3 for the line elements are expressed by the quadratic polynomials
and the conditions that must be met by the shape functions are
Thus, the following shape functions can be obtained for the secondorder line element:
Next, we calculate the matrix elements shown in Eqs. (7.29)-(7.32).
7.3
FINITE-ELEMENT ANALYSIS OF TIME-INDEPENDENT STATE
FIGURE 7.4. Second-order line element. /. I (d[Ne]/dx)(d[Ne]T/dx)
and therefore,
Since the relations
hold, we get
dx
From Eq. (7.49), we get
261
262
SCHRODINGER EQUATION
Thus, we get
This equation is summarized as
2- le [Ne\[Ne]T dx n
Through a similar procedure, we get
REFERENCES
263
Since Eqs. (7.63) and (7.64) can be used to construct the matrixes [P], [Q], and [R], the eigenenergy can be calculated from the eigenvalue matrix equation (7.35).
REFERENCES [1] K. Kawano, S. Sekine, H. Takeuchi, M. Wada, M. Kohtoku, N. Yoshimoto, T. Ito, M. Yanagibashi, S. Kondo, and Y. Noguchi, "4 x 4 InGaAlAs/InAlAs MQW directional coupler waveguide switch modules integrated with spot-size converters and their lOGbit/s operation," Electron. Lett., vol. 31, pp. 96-97, 1995. [2] K. Kawano, K. Wakita, O. Mitomi, I. Kotaka, and M. Naganuma, "Design of InGaAs/InAlAs multiple-quantum well (MQW) optical modulators," IEEEJ. Quantum Electron., vol. QE-28, pp. 228-230, 1992. [3] L. I. Schiff, Quantum Mechanics, McGraw-Hill, New York, 1968. [4] K. Nakamura, A. Shimizu, M. Koshiba, and K. Hayata, "Finite-element analysis of quantum wells of arbitray semiconductors with arbitrary potential potential profiles," IEEEJ. Quantum Electron., vol. 25, pp. 889-895, 1989. [5] M. Koshiba, H. Saitoh, M. Eguchi, and K. Hirayama, 'Simple scalar finiteelement approach to optical waveguides," IEE Proc. J., vol. 139, pp. 166171, 1992.
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APPENDIX A
VECTORIAL FORMULAS
In the following, i, j and k are respectively unit vectors in the x, y, and z directions and and A are respectively a scalar and a vector:
265
266
VECTORIAL FORMULAS
If r, 0, and z are respectively unit vectors in the radial, azimuthal, and longitudinal directions, the rotation formula for a vector A = Arr + AQ& + Azz is expressed as
A Laplacian V2 for a cylindrical coordinate is given as
APPENDIX B
INTEGRATION FORMULA FOR AREA COORDINATES
The integration formula shown in Eq. (3.184) is derived here by calculating the following integration for a triangular element e shown in Fig. 3.4:
where i,j, and k are integers and the spatial coordinates of nodes 1,2, and 3 are respectively (xi,yi\ (x2,y2), and (x3,y^). As shown in Eq. (3.72), we have the following relation between the soatial coordinates and the area coordinates:
As shown by the bottom row of Eq. (B.2),
268
INTEGRATION FORMULA FOR AREA COORDINATES
According to Eq. (B.2), x and y can be expressed using Ll and L2 as follows:
Thus, we get
where Se is the area of element e. Using Eq. (B.6) to transform x andy to Zq and L2, we can rewrite Eq. (B.I) as follows:
INTEGRATION FORMULA FOR AREA COORDINATES
The second integral of Eq. (B.7) is calculated as
where
Substituting Eq. (B.9) into Eq. (B.8), we get
269
270
INTEGRATION FORMULA FOR AREA COORDINATES
And substituting Eq. (B.10) into Eq. (B.7), we get
The remaining calculation is
Substituting I{(i +j + k+l, 0), where /j(i +y + k + 1, 0) can be rewitten as
into Eq. (B.I2), we get
INTEGRATION FORMULA FOR AREA COORDINATES
271
Then substituting this equation into Eq. (B.ll), we finally get the formula
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INDEX
Area coordinate, 74 Alternate-direction implicit (ADI) method, 216 Angular frequency, 3 Bandwidth, 109 Basis function, 62 Beam propagation method (BPM), 165 ADI-BPM, 216 first Fourier transform beam propagation method (FFT-BPM), 165 finite-difference beam propagation method (FD-BPM), 180 Bessel function, 40 Bessel function of first kind, 40 Bessel function of second kind, 40 modified Bessel function of first kind, 40 modified Bessel function of second kind, 40 Boundary condition, 9, 27, 110, 153, 197 absorbing boundary condition (ABC), 241 analytical boundary condition, 154 transparent boundary condition (TBC), 197
Characteristic equation, 17, 20, 41, 53 Charge density, 1 Cladding, 13, 37, 125 Core, 13, 37, 125 Cramer's formula, 76 Crank-Nicolson scheme, 195 Current density, 1 Cylindrical coordinate system, 38 Derivative, 119 first derivative, 119 second derivative, 119 Difference center, 131 hypothetical difference center, 131 Discretization, 130 equidistant discretization, 130 nonequidistant discretization, 130 Dirichlet condition, 66, 111, 153, 257 Dominant mode, 111 Effective index, 6 effective index method, 20 Eigenvalue, 88, 151 eigenvalue matrix equation, 68, 72, 151, 257 Eigenvector, 88, 151 Electric field, 1 273
274
INDEX
Electric flux density, 1 Element, 64 triangular element, 64, 72 first-order triangular element, 64, 73, 91, 106 second-order triangular element, 64, 79, 95, 108 ^ mode, 24, 113 Epq mode, 31, 113 Even mode, 111 Expansion coefficient, 63 Explicit scheme, 239 Finite-element method (FEM), 59 scalar finite-element method (SC-FEM), 59 Finite-difference method (FDM), 116 scalar finite-difference method (SC-FDM), 150 semivectorial finite-difference method (SV-FDM), 117 Finite-difference time-domain method (FD-TDM), 233 Fourier transform, 170 discrete Fourier transform, 170 inverse discrete Fourier transform, 170 Fresnel approximation, 167-168, 187 Functional, 62 Fully vectorial analysis, 222
Magnetic field, 1 Magnetic flux density, 1 Marcatili's method, 23 Maxwell's equations, 1 Matrix element, 89 Mirror-symmetrical plane, 111 Multistep method, 213 Neumann condition, 66, 111, 153, 257 Node, 64, 73, 79, 129, 257 Normalized frequency, 41 Odd mode, 111 Optical fiber, 36 step-index optical fiber, 37 Fade approximant operator, 204 Para-axial approximation, 167 Permeability, 1 relative permeability, 1 Permittivity, 1 relative permittivity, 1 Phase-shift lens, 170, 173 Phaser expression, 3 Plane wave, 10 Plank constant, 252 Potential, 252 Power confinement factor (r factor), 55-56 Poynting vector, 7 Principal field component, 125, 182, 184 Propagation constant, 6
Galerkin method, 68 Helmholtz equaiotn, 5-7 Hybrid-mode analysis, 47 Implicit scheme, 186 Interpolation function, 64 Joule heating, 8 Laplacian, 4 Line element, 257 first-order line element, 257 second-order line element, 260 Local coordinate, 107, 110 LP mode, 38
Quantum well, 252 Quasi-TE mode, 125, 128 Quasi-TM mode, 125, 147 Rayleigh-Ritz method, 62 Reference index, 166, 187 Residual, 68 error residual, 68 weighted residual method, 69 Schrodinger equation, 251 normalized Schrodinger equation, 255 time-dependent Schrodinger equation, 251
INDEX
time-independent Schrodinger equation, 253-254 Shape function, 64, 78, 83 Slab optical waveguide, 13 Slowly varying envelope approximation (SVEA), 166 Stability condition, 195, 239 Taylor series expansion, 118 Tohmas method, 203 Transverse electric (TE) mode, 14, 181, 186 Transverse magnetic (TM) mode, 14, 184, 190
Variational method, 59 Variational principle, 62 Wave equation, 5, 6 scalar wave equation, 84, 127 semivectorial wave equation, 124 vectorial wave equation, 4, 120 Wave number, 5 Weak form, 69 Wide-angle formulation, 167 Wide-angle analysis, 204 Wide-angle order, 205 Yee lattice, 235
275
